University of Florida/Egm4313/s12.team11.R3

Intermediate Engineering Analysis Section 7566 Team 11 Due date: February 22, 2012.

Problem 3.1
Solved by Luca Imponenti

Problem Statement
Find the solution to the following L2-ODE-CC: $$y''-10y'+25y=r(x)\!$$

With the following excitation: $$r(x)=7e^{5x}-2x^2\!$$

And the following initial conditions: $$y(0)=4, y'(0)=-5\!$$

Plot this solution and the solution in the example on p.7-3

Homogeneous Solution
To find the homogeneous solution we need to find the roots of our equation

$$\lambda^2-10\lambda+25=0\!$$ $$(\lambda-5)(\lambda-5)=0\!$$ $$\lambda=5\!$$

We know the homogeneous solution for the case of a real double root with $$\lambda=5\!$$ to be

$$y_h=c_1e^{5x}+c_2xe^{5x}\!$$

Particular Solution
For the given excitation we must use the Sum Rule to the particular solution as follows

$$y_p=y_{p1}+y_{p2}\!$$ where $$y_{p1}\!$$ and $$y_{p2}\!$$ are the solutions to $$r_1(x)=7e^{5x}\!$$ and $$r_2(x)=-2x^2\!$$, respectively

First Particular Solution
$$r_1(x)=7e^{5x}\!$$,

from table 2.1, K 2011, pg. 82 we have

$$y_{p1}=Ce^{5x}\!$$

but this corresponds to one of our homogeneous solutions so we must use the modification rule to get

$$y_{p1}=Cx^2e^{5x}\!$$

Plugging this into the original L2-ODE-CC then substituting;

$$y_{p1}''-10y_{p1}'+25y_{p1}=r_1(x)\!$$

$$(Cx^2e^{5x})''-10(Cx^2e^{5x})'+25(Cx^2e^{5x})=r_1(x)\!$$

$$25Cx^2e^{5x}+10Cxe^{5x}+10Cxe^{5x}+2Ce^{5x}-10(5Cx^2e^{5x}+2Cxe^{5x})+25Cx^2e^{5x}=7e^{5x}\!$$

$$e^{5x}[25Cx^2+10Cx+10Cx+2C-50Cx^2-20Cx+25Cx^2]=7e^{5x}\!$$

$$e^{5x}2C=7e^{5x}\!$$

so $$C=\frac{7}{2}\!$$ and the first particular solution is,
 * $$y_{p1}=\frac{7}{2}x^2e^{5x}\!$$

Second Particular Solution
$$r_2(x)=-2x^2\!$$,

from table 2.1, K 2011, pg. 82 we have

$$y_{p2}=a_2x^2+a_1x+a_0\!$$

Plugging this into the original L2-ODE-CC then substituting;

$$y_{p2}''-10y_{p2}'+25y_{p2}=r_2(x)\!$$

$$(a_2x^2+a_1x+a_0)''-10(a_2x^2+a_1x+a_0)'+25(a_2x^2+a_1x+a_0)=-2x^2\!$$

$$2a_2-10(2a_2x+a_1)+25(a_2x^2+a_1x+a_0)=-2x^2\!$$

grouping like terms we get three equations to solve for the three unknowns, these are written in matrix form

$$25a_2x^2+(25a_1-20a_2)x+(25a_0-10a_1+2a_2)=-2x^2\!$$


 * $$ \begin{bmatrix}

2 & -10 & 25\\ -20 & 25 & 0\\  25 & 0 & 0\end{bmatrix}

\begin{bmatrix} a_2\\ a_1\\ a_0\end{bmatrix} = \begin{bmatrix} 0\\ 0\\ -2\end{bmatrix}\!$$

solving by back subsitution leads to $$a_2=-\frac{2}{25}, a_1=\frac{8}{125}, a_0=\frac{4}{125}\!$$

so the second particular solution is,
 * $$y_{p2}=-\frac{2}{25}x^2+\frac{8}{125}x+\frac{4}{125}\!$$

General Solution
The general solution is the summation of the homogeneous and particular solutions

$$y=y_h+y_{p1}+y_{p2}\!$$

$$y=c_1e^{5x}+c_2xe^{5x}+\frac{7}{2}x^2e^{5x}-\frac{2}{25}x^2+\frac{8}{125}x+\frac{4}{125}\!$$

$$y=e^{5x}(c_1+c_2x+\frac{7}{2}x^2)-\frac{2}{25}x^2+\frac{8}{125}x+\frac{4}{125}\!$$

Applying the first initial condition $$y(0)=4\!$$

$$y(0)=c_1+\frac{4}{125}=4\!$$

$$c_1=\frac{496}{125}\!$$

Second initial condition $$y'(0)=-5\!$$

$$y'=\frac{d}{dx}y=e^{5x}[5(c_1+c_2x+\frac{7}{2}x^2)+c_2+7x]-\frac{4}{25}x+\frac{8}{125}\!$$

$$y'=e^{5x}[\frac{35}{2}x^2+(5c_2+7)x+5c_1+c_2]-\frac{4}{25}x+\frac{8}{125}\!$$

$$y'(0)=5c_1+c_2+\frac{8}{125}=-5\!$$

$$c_2=-\frac{3113}{125}\!$$

The general solution to the differential equation is therefore

$$y(x)=e^{5x}(\frac{496}{125}-\frac{3113}{125}x+\frac{7}{2}x^2)-\frac{2}{25}x^2+\frac{8}{125}x+\frac{4}{125}\!$$

Plot
Below is a plot of this solution and the solution to in the example on p.7-3

our solution $$y(x)=e^{5x}(\frac{496}{125}-\frac{3113}{125}x+\frac{7}{2}x^2)-\frac{2}{25}x^2+\frac{8}{125}x+\frac{4}{125}\!$$ (shown in red)

example on p.7-3 $$y(x)=e^{5x}(4-25*x+*\frac{7}{2}x^2)\!$$ (shown in blue)



Egm4313.s12.team11.imponenti 05:55, 20 February 2012 (UTC)

Problem 3.2
Solved by Gonzalo Perez

Problem Statement
Developing the second homogeneous solution for the case of double real root as a limiting case of distinct roots.

Given
Consider two distinct roots of the form:

$$ \lambda_1 = x \! $$ and $$ \lambda_2 = x + \epsilon \! $$

(where $$ \epsilon \! $$ is perturbation).

Given
Find the homogeneous L2-ODE-CC having the above distinct roots.

Solution
$$ (\lambda - \lambda_1)(\lambda - (\lambda_1))=0 \! $$

$$ (\lambda - x)(\lambda - (x + \epsilon))=0 \! $$

$$ \lambda^2 - \lambda x - \lambda \epsilon - \lambda x + x^2 + x \epsilon =0 \! $$

$$ \lambda^2 - \lambda (2x+\epsilon) + x (x+\epsilon )=0 \! $$

$$ \therefore y'' - y'(2\lambda +\epsilon) + y \lambda (\lambda + \epsilon) = 0 \! $$ (1)

Given
Show that $$ \frac{e^{(\lambda +\epsilon) x} - e^{\lambda x}}{\epsilon } \! $$ is a homogeneous solution. (2)

Solution
Let's find the corresponding derivatives:

$$ y = \frac{e^{(\lambda+\epsilon) x} - e^{\lambda x}}{\epsilon} \! $$

$$ y' = \frac{(\lambda +\epsilon) e^{\lambda + \epsilon x}-\lambda e^{\lambda x}}{\epsilon } \! $$

$$ y'' = \frac{(\lambda +\epsilon)^2 e^{(\lambda + \epsilon) x}-\lambda^2 e^{\lambda x}}{\epsilon} \! $$

If we now take these three equations and plug them into the homogeneous L2-ODE-CC (1), we get:

$$ e^{(\lambda + \epsilon)x} (\lambda^2+2\lambda \epsilon + \epsilon^2 - 2\lambda^2-2\lambda \epsilon - \lambda \epsilon -\epsilon^2 + \lambda^2 + \epsilon \lambda)+e^{\lambda x}(- \lambda^2+2\lambda^2+\lambda \epsilon - \lambda ^2 - \lambda \epsilon) = 0 \! $$

$$ \therefore 0 \equiv 0. \! $$

Since the left and right hand sides of the equation are zero, the solution is in fact a homogeneous equation.

Given
Find the limit of the homogeneous solution in (2) as epsilon approaches zero (think l'Hopital's Rule).

Solution
Using l'Hopital's Rule,

$$ \lim_{\epsilon \rightarrow 0}\frac{e^{(\lambda +\epsilon) x }-e^{\lambda x}}{\epsilon} = \frac{0}{0} \! $$

(this is an indeterminate form).

L'Hopital's Rule states that we can divide this function into two functions, $$ f(\epsilon) \! $$ and $$ g(\epsilon) \! $$, and then find their derivatives and attempt to find the limit of $$ \frac{f'(\epsilon)}{g'(\epsilon)} \! $$. If a limit exists for this, then a limit exists for our original function.

$$ \lim_{\epsilon \rightarrow 0} \frac{f'(\epsilon)}{g'(\epsilon)} = \lim_{\epsilon \rightarrow 0} \frac{xe^{x(\lambda + \epsilon)}}{1}\! $$

$$ = \frac{x e^{(\lambda+0)x}}{1} \! $$

$$ = xe^{x \lambda} \! $$

Given
Take the derivative of $$ e^{\lambda x} $$ with respect to lambda.

Solution
Taking the derivative with respect to lambda, we find that:

$$ \frac{d(e^{\lambda x})}{d\lambda} = xe^{\lambda x} $$.

It is important to remember that we must hold $$ x \! $$ as a constant when finding this derivative.

Given
Compare the results in parts (3) and (4), and relate to the result by using variation of parameters

Solution
Taking a closer look at Parts 3 and 4 of this problem, we discover that they're in fact equal:

$$ \frac{d(e^{\lambda x})}{d\lambda }=\lim_{e\rightarrow 0}\frac{e^{(\lambda +\epsilon )x} - e^{\lambda x}} {\epsilon} = xe^{\lambda x} \! $$

Given
Numerical experiment: Compute (2) setting lambda equal to 5 and epsilon equal to 0.001, and compare to the value obtained from the exact second homogeneous solution.

Solution
After performing these calculations, from (2) we get 148.478.

And from the exact second homogeneous solution, we get 200.05.

Egm4313.s12.team11.perez.gp 20:32, 22 February 2012 (UTC)

Problem 3.3
Solved by Jonathan Sheider

Problem Statement
Find the complete solution to the ODE with the given initial conditions. Plot the solution y(x).

Given
$${y}''-3{y}'+2y = 4x^2\!$$ $$y(0) = 0, {y}'(0) = 0\!$$

Solution
First let us analyze the homogenous solution to the given ODE: $${y}''-3{y}'+2y = 0\!$$ The characteristic equation for this ODE is therefore: $$\lambda^2 - 3\lambda + 2 = 0\!$$ Solving for $$\lambda\!$$: $$(\lambda-2)(\lambda-1) = 0\!$$ $$\lambda = {1,2}\!$$ $$\lambda_{1} = 1\!$$ and $$\lambda_{2} = 2\!$$ Therefore the equation has two real roots and a homogenous solution of the following form (from Kreyszig 2011, p.54-57): $$ y_{h} = c_{1}e^{\lambda_{1}x} + c_{2}e^{\lambda_{2}x}\!$$ And finally we find the general homogenous solution: $$ y_{h} = c_{1}e^{x} + c_{2}e^{2x}\!$$ Next let us evaluate the particular solution to the given ODE: $${y}''-3{y}'+2y = 4x^2\!$$ The function on the right hand side of this equation implies that the particular solution has the following form based on Table 2.1 (from Kreyszig 2011, p.82): $$ y_{p} = K_{n}x^n + K_{n-1}x^{n-1} + ... + K_{1}x + K_{0} \!$$ Therefore for this ODE we have: $$ y_{p} = K_{2}x^2+K_{1}x+K_{0}\!$$ Differentiating $$ y_{p}\!$$ to obtain $${y_{p}}'\!$$ and $${y_{p}}''\!$$ respectively: $$ {y}' = 2K_{2}x+K_{1}\!$$ $$ {y}'' = 2K_{2}\!$$ Substituting these equations into the original ODE yields: $$ (2K_{2}) -3(2K_{2}x+K_{1}) + 2(K_{2}x^2+K_{1}x+K_{0}) = 4x^2\!$$ $$ 2K_{2} -6K_{2}x-3K_{1} + 2K_{2}x^2+2K_{1}x+2K_{0} = 4x^2\!$$ And we know that the coefficients of the variables on each side of the equation must be equal: $$ (2K_{2})x^2 + (2K_{1}-6K_{2})x + (2K_{2} - 3K_{1} + 2K_{0}) = 4x^2\!$$ Therefore we find: $$ 2K_{2} = 4\!$$ $$ K_{2} = 2 \!$$ Now, solving for $$ K_{1}\!$$ and $$ K_{0}\!$$: $$ 2K_{1} - 6K_{2} = 0 \!$$ $$ 2K_{1} = 6(2) \!$$ $$ K_{1} = 6\!$$ And also: $$ 2K_{2} - 3K_{1} + 2K_{0} = 0 \!$$ $$ 2K_{0} = 3(6) - 2(2)\!$$ $$ K_{0} = 7\!$$ Finally we arrive at the particular solution: $$ y_{p} = 2x^2 + 6x + 7\!$$ By superposition, we can find the complete general solution: $$ y = y_{h} + y_{p}\!$$ $$ y = c_{1}e^{x} + c_{2}e^{2x} + 2x^2 + 6x + 7\!$$ Using the given initial conditions, we can solve for $$ c_{1}\!$$ and $$c_{2}\!$$, first by using the initial conditions for $$y\!$$: $$ y(0) = c_{1} + c_{2} + 7 = 0\!$$ Differentiating the complete general solution and using the given initial condition for $${y}'\!$$: $$ {y}' = c_{1}e^{x} + 2c_{2}e^{2x} + 4x + 6\!$$ $$ {y}'(0) = c_{1} + 2c_{2} + 6 = 0\!$$ Solving this system of equations, by solving the first equation for $$c_{1}\!$$: $$c_{1} = -7 - c_{2}\!$$ Plugging this into the second equation yields: $$ (-7 - c_{2}) +2c_{2} + 6 = -1 + c_{2} = 0\!$$ $$ c_{2} = 1\!$$ And therefore: $$ c_{1} = -7 - (1)\!$$ $$ c_{1} = -8\!$$ Finally we have the complete general solution that evaluates the given initial conditions: $$ y(x) = -8e^{x} + e^{2x} +2x^2+6x+7\!$$ A plot of $$y(x)\!$$ from $$x = -3\!$$ to $$x = 3\!$$ is shown using MatLAB:



Checking
Differentiating the solution to find $$ {y}'\!$$ and $$ {y}''\!$$ respectively: $$ y = -8e^{x} + e^{2x} +2x^2+6x+7\!$$ $$ {y}' = -8e^{x} + 2e^{2x} +4x+6\!$$ $$ {y}'' = -8e^{x} + 4e^{2x} +4\!$$ Substituting into the original ODE yields: $$ {y}'' - 3{y}'+2y = 4x^2\!$$ $$ (-8e^{x} + 4e^{2x} +4) -3(-8e^{x} + 2e^{2x} +4x+6) +2(-8e^{x} + e^{2x} +2x^2+6x+7) = 4x^2\!$$ $$ -8e^{x} + 4e^{2x} +4 +24e^{x} - 6e^{2x} -12x-18 -16e^{x} + 2e^{2x} +4x^2+12x+14 = 4x^2\!$$ $$ (-8+24-16)e^{x} + (4-6+2)e^{2x} + (-12+12)x + (-18+4+14) +4x^2 = 4x^2\!$$ $$ 4x^2 = 4x^2\!$$ Therefore this solution is correct. --Egm4313.s12.team11.sheider 21:30, 21 February 2012 (UTC)

Problem 3.4
Solved by Daniel Suh

Problem Statement
Use Basic Rule (1) and Sum Rule (3) to show that the appropriate particular solution for $$y''-3y'+2y = 4x^{2}-6x^{5}\!$$ is of the form $$\sum_{j=0}^{n}c_{j}x^{j}\!$$, with n = 5.

Finding the Particular Solution with Basic Rule and Sum Rule
We know that in standard form, for a particular solution, $$y''_{p}+ay'_{p}+by_{p}=r(x)\!$$ $$y_{p}=y_{p1}+y_{p2}\!$$

Using Basic Rule (1) and Sum Rule (3), we know that $$y''_{p}+ay'_{p}+by_{p}=\sum_{i}r_i(x)\!$$

Additionally, we choose $$r_1(x)\rightarrow y_{p1}(x)\!$$ $$r_2(x)\rightarrow y_{p2}(x)\!$$

Solving for Particular Solution 1
Using the Method of Undetermined Coefficients, we find that $$y_{p1}=C_{2}x^{2}+C_{1}x+C_{0}\!$$ $$y'_{p1}=2C_{2}x+C_{1}\!$$ $$y''_{p1}=2C_{2}\!$$

Plugging $$y_{p1}\!$$ into $$y''-3y'+2y=4x^{2}\!$$ gives $$(2C_{2})-3(2C_{2}x+C_{1})+2(C_{2}x^{2}+C_{1}x+C_{0})=4x^{2}\!$$

Solving for coefficients $$2C_{2}=4\!$$ $$-6C_{2}+2C_{1}=0\!$$ $$-3C_{1}+2C_{0}+2C_{2}=0\!$$

Results in $$C_{2}=2\!$$ $$C_{1}=6\!$$ $$C_{0}=7\!$$

$$y_{p1}=2x^{2}+6x+7\!$$

Solving for Particular Solution 2
Using the Method of Undetermined Coefficients, we find that $$y_{p2}=K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+K_{2}x^{2}+K_{1}x+K_{0}\!$$ $$y'_{p2}=5K_{5}x^{4}+4K_{4}x^{3}+3K_{3}x^{2}+2K_{2}x+K_{1}\!$$ $$y''_{p2}=20K_{5}x^{3}+12K_{4}x^{2}+6K_{3}x+2K_{2}\!$$

Plugging $$y_{p2}\!$$ into $$y''-3y'+2y=-6x^{5}\!$$ gives $$(20K_{5}x^{3}+12K_{4}x^{2}+6K_{3}x+2K_{2})-3(5K_{5}x^{4}+4K_{4}x^{3}+3K_{3}x^{2}+2K_{2}x+K_{1})+2(K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+K_{2}x^{2}+K_{1}x+K_{0})=-6x^{5}\!$$

Solving for coefficients results in $$K_{5}=-3\!$$ $$K_{4}=-22.5\!$$ $$K_{3}=-105\!$$ $$K_{2}=-337.5\!$$ $$K_{1}=-697.5\!$$ $$K_{0}=-708.75\!$$

$$y_{p2}=-3x^{5}-22.5x^{4}-105x^{3}-337.5x^{2}-697.5x-708.75\!$$

Combining Particular Solutions
Plugging $$y_{p1}\!$$ and $$y_{p2}\!$$ into $$y_{p}=y_{p1}+y_{p2}\!$$ gives

$$y_{p}=-3x^{5}-22.5x^{4}-105x^{3}-335.5x^{2}-691.5x-701.75\!$$

Comparing to Summation Form
The particular solution in the summation form is $$y_{p}=\sum_{j=0}^{n}c_{j}x^j\!$$

if n=5, then $$y_{p}=\sum_{j=0}^{5}c_{j}x^j\!$$ $$y'_{p}=\sum_{j=0}^{5}c_{j}*j*x^{(j-1)}\!$$ $$y''_{p}=\sum_{j=0}^{5}c_{j}*j*(j-1)*x^{(j-2)}\!$$

Plugging the particular solution into $$y''-3y'+2y=-6x^{5}\!$$, the final solution gives $$y_{p}=-3x^{5}-22.5x^{4}-105x^{3}-335.5x^{2}-691.5x-701.75\!$$ (check Problem 3.5 for explanation on how to get this answer)

Result
As you can see, the two particular solutions of $$y_{p}\!$$ are equal. Thus, using the Basic Rule (1) and Sum rule (3) does give you the correct particular solution for $$y''-3y'+2y=4x^{2}-6x^{5}\!$$, in the form of $$y_{p}=\sum_{j=0}^{n}c_{j}x^j\!$$

Created by [| Daniel Suh] 20:57, 21 February 2012 (UTC)

Problem 3.5
Solved by: Andrea Vargas

Problem Statement
Given $$y''-3y'+2y=4x^2-6x^5\!$$

1. Obtain the coefficients of $$ x,x^2,x^3,x^5 \!$$ 2.Verify all equations by long-hand expansion. Use the series before adjusting the indexes.

3. Put the system of equations in an upper triangular matrix.

4. Solve for $$c_0,c_1,c_2,c_3,c_4,c_5\!$$ by using back substitution.

5. Using the initial conditions $$ y(0)=1, y'(0)=0 \!$$ find $$y(x)\!$$ and plot it

Solution
1. To obtain the equations for the coefficients we use the following equation from (1) p7-11:

$$\sum_{j=0}^{3}[c_{j+2}(j+2)(j+1)-3c_{j+1}(j+1)+2c_j]x^j-3c_5(5)x^4+2[c_4x^4+c^5x^5]=4x^2-6x^5 \!$$

Finding the coefficients of $$x\!$$ where $$j=1\!$$:

$$[c_3(3)(2)-3(c_2)(2)+2c_1]x^1=[6c_3-6c_2+2c_1]x\!$$

Finding the coefficients of $$x^2\!$$ where $$j=2\!$$:

$$[c_4(4)(3)-3(c_3)(3)+2c_2]x^2=[12c_4-9c_3+2c_2]x^2\!$$

Finding the coefficients of $$x^3\!$$ where $$j=3\!$$:

$$[c_5(5)(4)-3(c_4)(4)+2c_3]x^3=[20c_4-12c_3+2c_2]x^3\!$$

Finding the coefficients of $$x^5\!$$ where $$j=5\!$$:

$$c_5x^5\!$$

2. To verify all the above equations by long hand expansion we use the following equation from (4) p7-12: $$\sum_{j=2}^{5}c_j\times j\times(j-1)\times x^{j-2}-3\sum_{j=1}^5c_j\times j\times x^{j-1}+2\sum_{j=0}^5c_j\times x^j=4x^2-6x^5\!$$

Finding the coefficients when $$j=0\!$$:

$$2c_0\!$$

Finding the coefficients when $$j=1\!$$:

$$-3c_1+2c_1x_1\!$$

Finding the coefficients when $$j=2\!$$:

$$2c_2-6c_2x+2c_2x^2\!$$

Finding the coefficients when $$j=3\!$$:

$$6c_3x-9c_3x^2+2c_3x^3\!$$

Finding the coefficients when $$j=5\!$$:

$$20c_5x^3-15c_5x^4+2c_5x^5\!$$

By collecting these terms we can compare them to the equations of part 1.

Coefficients of $$x\!$$:

$$[c_3(3)(2)-3(c_2)(2)+2c_1]x^1=[6c_3-6c_2+2c_1]x\!$$

Coefficients of $$x^2\!$$:

$$[c_4(4)(3)-3(c_3)(3)+2c_2]x^2=[12c_4-9c_3+2c_2]x^2\!$$

Coefficients of $$x^3\!$$:

$$[c_5(5)(4)-3(c_4)(4)+2c_3]x^3=[20c_4-12c_3+2c_2]x^3\!$$

Coefficients of $$x^5\!$$:

$$c_5x^5\!$$

We can see that we obtain the same system of equations to solve for the coefficients with both methods.

3.Constructing the coefficients matrix: $$\begin{bmatrix} 2&-3&2&0&0&0 \\ 0&2&-6&6&0&0 \\  0&0&2&-9&12&0 \\  0&0&0&2&-12&20 \\  0&0&0&0&2&-15 \\  0&0&0&0&0&2 \end{bmatrix}\!$$

Then, the system becomes:

$$\begin{bmatrix} 2&-3&2&0&0&0 \\ 0&2&-6&6&0&0 \\  0&0&2&-9&12&0 \\  0&0&0&2&-12&20 \\  0&0&0&0&2&-15 \\  0&0&0&0&0&2 \end{bmatrix} \begin{bmatrix} c_0 \\ c_1 \\ c_2 \\ c_3 \\ c_4 \\ c_5 \\ \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 4 \\ 0 \\ 0 \\ -6 \\ \end{bmatrix}\!$$ 4. Solving for the coefficients: $$2c_5=-6 \rightarrow c_5=-3\!$$ $$ 2c_4-(15)(-3)=0\rightarrow c_4=\frac{-45}{2}\!$$ $$ 2c_3-12(-\frac{45}{2})+20(-3)=0 \rightarrow c_3=-105 \!$$ $$ 2c_2-9(-105)+12(-\frac{45}{2})=4 \rightarrow c_2=-\frac{671}{2}\!$$ $$ 2c_1-6(-\frac{671}{2})+6(-105)=0 \rightarrow c_1=-\frac{1383}{2} \!$$ $$ 2c_0-3(-\frac{1383}{2})+2(-\frac{671}{2})=0 \rightarrow c_0=-\frac{2807}{4} \!$$

Particular solution
This yields the particular solution: $$y_p(x)=-3x^5-\frac{45}{2}x^4-105x^3-\frac{671}{2}x^2-\frac{1383}{2}x-\frac{2807}{4}\!$$

Homogeneous Solution
$$y{}''_h-y{}'_h+2y_h=0\!$$ $$\lambda^2-3\lambda+2=0\!$$ Then, we can find the characteristic equation: $$(\lambda-2)(\lambda-1)\!$$ $$\lambda=2,1\!$$ Then the solution for the homogeneous equation becomes: $$y_h(x)=C_1e^{2x}+C_2e^{x}\!$$

General Solution
Using the given initial conditions $$ y(0)=1, y'(0)=0 \!$$ we find the overall solution:

$$ y(x)=y_h+y_p \!$$ $$ y(x)=C_1e^{2x}+C_2e^x-3x^5-\frac{45}{2}x^4-105x^3-\frac{671}{2}x^2-\frac{1383}{2}x-\frac{2807}{4} \!$$ $$ y'(x)=2C_1e^{2x}+C_2e^x-15x^4-90x^3-315x^2-671x-\frac{1383}{2}\!$$ Using the initial conditions to solve for $$ C_1 \!$$ and $$ C_2 \!$$ $$ 1=C_1+C_2-\frac{2807}{4} \!$$ $$ 0=C_1+C_2-\frac{1383}{2} \!$$ $$ C_1=-\frac{45}{4} \; \; \; C_2=714 \!$$ The general solution becomes

$$ y(x)=-\frac{45}{4}e^{2x}+714e^x-3x^5-\frac{45}{2}x^4-105x^3-\frac{671}{2}x^2-\frac{1383}{2}x-\frac{2807}{4} \!$$

Plot
Below is a plot of the solution:



--Egm4313.s12.team11.vargas.aa 05:56, 21 February 2012 (UTC)

Problem 3.6
Solved by Francisco Arrieta

Problem Statement

 * Solve the L2-ODE-CC (2) p.7-11 with initial conditions (2b) p.3-6 differently as follows. Consider the following two L2-ODE-CC (see p. 7-2b)


 * $$ y_{p,1}''-3y_{p,1}'+2y_{p,1}=r_1(x)=4x^2 \!$$


 * $$ y_{p,2}''-3y_{p,2}'+2y_{p,2}=r_2(x)=-6x^5 \!$$


 * The particular solution to $$ y_{p,1} \!$$ had been found in R3.3 p.7-11.
 * Find the particular solution $$ y_{p,2} \!$$, and then obtain the solution $$ y \!$$ for the L2-ODE-CC (2) p.7-11 with initial conditions (2b) p.3-6.

Given

 * First particular solution


 * $$ y_{p,1}=2x^2+6x+7 \!$$


 * Initial Conditions


 * $$ y(0)=1 \!$$
 * $$ y'(0)=0 \!$$

Particular Solution

 * Since the specific excitation $$ r_2(x)=-6x^5 \!$$, using table 2.1 from K 2011 p.82, the choice for the particular solution is $$ y_p(x)=\sum_{j=0}^{n}K_jx^j \!$$


 * Then


 * $$ y_{p,2}=c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5 \!$$
 * $$ y_{p,2}'=c_1+2c_2x+3c_3x^2+4c_4x^3+5c_5x^4 \!$$
 * $$ y_{p,2}''=2c_2+6c_3x+12c_4x^2+20c_5x^3 \!$$


 * Plugging these equations back in the original $$ y_{p,2}''-3y_{p,2}'+2y_{p,2}=-6x^5 \!$$


 * $$ (2c_2+6c_3x+12c_4x^2+20c_5x^3)-3(c_1+2c_2x+3c_3x^2+4c_4x^3+5c_5x^4)+2(c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5)=-6x^5 \!$$


 * Using coefficient matching of left hand side to the right hand side of the equation, the following equations are obtained


 * $$ c:  2c_2-3c_1+2c_0=0 \!$$
 * $$ x:  6c_3-6c_2+2c_1=0 \!$$
 * $$ x^2:  12c_4-9c_3+2c_2=0 \!$$
 * $$ x^3:  20c_5-12c_4+2c_3=0 \!$$
 * $$ x^4:  -15c_5+2c_4=0 \!$$
 * $$ x^5:  2c_5=-6 \!$$


 * Which is equivalent to the following matrix


 * $$ \begin{bmatrix}

2&-3 & 2 & 0 & 0 & 0\\ 0& 2 &-6 & 6 & 0 & 0\\  0& 0 & 2 &-9 &12 & 0\\  0& 0 & 0 & 2 &-12& 20\\  0& 0 & 0 & 0 & 2 & -15\\  0& 0 & 0 & 0 & 0 & 2 \end{bmatrix}

\begin{bmatrix} c_0\\ c_1\\ c_2\\ c_3\\ c_4\\ c_5 \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ -6 \end{bmatrix} \!$$


 * Using back substitution method to solve for every coefficient, starting with $$ c_5 \!$$


 * $$ 2c_5=-6 \rightarrow c_5=-3 \!$$
 * $$ 2c_4-15(-3)=0 \rightarrow c_4=-\frac{45}{2} \!$$
 * $$ 2c_3-12(-\frac{45}{2})+20(-3)=0 \rightarrow c_3=-105 \!$$
 * $$ 2c_2-9(-105)+12(-\frac{45}{2})=0 \rightarrow c_2=-\frac{675}{2} \!$$
 * $$ 2c_1-6(-\frac{675}{2})+6(-105)=0 \rightarrow c_1=-\frac{1395}{2} \!$$
 * $$ 2c_0-3(-\frac{1395}{2})+2(-\frac{675}{2})=0 \rightarrow c_0=-\frac{2835}{4} \!$$


 * Plugging these values back into $$ y_{p,2}=c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5 \!$$ gives

$$ y_{p,2}=-\frac{2835}{4}-\frac{1395}{2}x-\frac{675}{2}x^2-105x^3-\frac{45}{2}x^4-3x^5 \!$$


 * Since these equations are L2-ODE-CC, the superposition principle applies and $$ y_p=y_{p,1}+y_{p,2} \!$$ and the general particular solution becomes

$$ y_p(x)=-\frac{2807}{4}-\frac{1383}{2}x-\frac{671}{2}x^2-105x^3-\frac{45}{2}x^4-3x^5 \!$$

Homogeneous Solution

 * $$ y_{h}''-3y_{h}'+2y_{h}=0 \!$$


 * Due to the linearity of this homogeneous equation, a linear combination of two linear independent solutions is also a solution


 * $$ y_{h,1}''-3y_{h,1}'+2y_{h,1}=0 \!$$
 * $$ y_{h,2}''-3y_{h,2}'+2y_{h,2}=0 \!$$


 * With solutions


 * $$ y_{h,1}=e^{\lambda_1x} \!$$
 * $$ y_{h,2}=e^{\lambda_2x} \!$$


 * In order to determine the value of $$ \lambda_{1,2} \!$$, the characteristic equation must be determine from the homogeneous equation


 * $$ \lambda^2-3\lambda+2=0 \!$$


 * $$ \lambda_{1,2}=\frac{-a\pm \sqrt{a^2-4b}}{2} \!$$
 * $$ \lambda_{1,2}=\frac{-(-3)\pm \sqrt{(-3)^2-4(2)}}{2} \!$$
 * $$ \lambda_1=2 \; \; \; \; \; \; \; \lambda_2=1\!$$


 * Then the solutions for each distinct linearly independent homogeneous equation becomes


 * $$ y_{h,1}=e^{2x} \!$$
 * $$ y_{h,2}=e^{x} \!$$


 * Because of linearity, the linear combination of the previous two equations times two constants that satisfy 2 initial conditions, is also a solution

$$ y_h(x)=C_1e^{2x}+C_2e^x \!$$

General Solution

 * The overall solution for the L2-ODE-CC


 * $$ y(x)=y_h+y_p \!$$
 * $$ y(x)=C_1e^{2x}+C_2e^x-\frac{2807}{4}-\frac{1383}{2}x-\frac{671}{2}x^2-105x^3-\frac{45}{2}x^4-3x^5 \!$$
 * $$ y'(x)=2C_1e^{2x}+C_2e^x-\frac{1383}{2}-671x-315x^2-90x^3-15x^4 \!$$


 * Using the initial conditions to solve for $$ C_1 \!$$ and $$ C_2 \!$$


 * $$ 1=C_1+C_2-\frac{2807}{4} \!$$
 * $$ 0=C_1+C_2-\frac{1383}{2} \!$$
 * $$ C_1=-\frac{45}{4} \; \; \; C_2=714 \!$$


 * The general solution becomes

$$ y(x)=-\frac{45}{4}e^{2x}+714e^x-\frac{2807}{4}-\frac{1383}{2}x-\frac{671}{2}x^2-105x^3-\frac{45}{2}x^4-3x^5 \!$$

--Egm4313.s12.team11.arrieta 20:26, 19 February 2012 (UTC)

Problem 3.7
Solved By Kyle Gooding

Problem Statement
Expand the series on both sides of (1),(2) pg. 7-12b to verify these equalities. (1)

Given
$$\sum_{j=2}^{5}C_j*j(j-1)x^{j-2}= \sum_{j=0}^{3}C_{j+2}*(j+2)(j+1)x^{j} $$

(2) $$ \sum_{j=1}^{5}C_j*jx^{j-1}= \sum_{j=0}^{4}C_{j+1}*(j+1)x^{j}$$

Solutions
Expanding both sides of (1) results in: $$\sum_{j=2}^{5}C_j*j(j-1)x^{j-2}=C_2(2)(2-1)x^0+ C_3(3)(3-1)x^1+C_4(4)(4-1)x^2 + C_5(5)(5-1)x^3 $$ $$\sum_{j=0}^{3}C_j(j+2)(j+1)x^{j}=C_{0+2}(0+2)(0+1)x^0+ C_{1+2}(1+2)(1+1)x^1+C_{2+2}(2+2)(2+1)x^2 + C_{3+2}(3+2)(3+1)x^3 $$

Simplifying: $$\sum_{j=2}^{5}C_j*j(j-1)x^{j-2}=C_2(2)+ C_36x+C_412x^2 + C_520x^3 $$ $$\sum_{j=0}^{3}C_j(j+2)(j+1)x^{j}=C_2(2)+ C_36x+C_412x^2 + C_520x^3 $$

The two sums are equal. Expanding both sides of (2) results in: $$ \sum_{j=1}^{5}C_j*jx^{j-1}= C_1(1)x^0+ C_2(2)x^1+C_3(3)x^2+C_4(4)x^3 + C_5(5)x^4 $$ $$ \sum_{j=0}^{4}C_{j+1}*(j+1)x^{j}= C_{0+1}(0+1)x^0+ C_{1+1}(1+1)x^1+C_{1+2}(2+1)x^2+C_{2+2}(3+1)x^3 + C_{3+2}(4+1)x^4 $$

Simplifying: $$ \sum_{j=1}^{5}C_j*jx^{j-1}= C_1(1)+ C_2(2)x^1+C_3(3)x^2+C_4(4)x^3 + C_5(5)x^4 $$ $$ \sum_{j=1}^{5}C_j*jx^{j-1}= C_1(1)+ C_2(2)x^1+C_3(3)x^2+C_4(4)x^3 + C_5(5)x^4 $$ The two sums are equal.

Egm4313.s12.team11.gooding 23:49, 19 February 2012 (UTC)

Problem 3.8
solved by Luca Imponenti

Problem Statement
Find a (real) general solution. State which rule you are using. Show each step of your work.
 * $$y''+4y'+4y=e^{-x}cos(x)\!$$

Homogeneous Solution
To find the homogeneous solution, $$y_h\!$$, we must find the roots of the equation

$$\lambda^2+4\lambda+4=0\!$$

$$(\lambda+2)(\lambda+2)=0\!$$

$$\lambda=-2\!$$

We know the homogeneous solution for the case of a double root to be

$$y_h=c_1e^{\lambda x}+c_2xe^{\lambda x}\!$$


 * $$y_h=c_1e^{-2x}+c_2xe^{-2x}\!$$

Particular Solution
We have the following excitation
 * $$r(x)=e^{-x}cos(x)\!$$

From table 2.1, K 2011, pg. 82, we have
 * $$y_p(x)=e^{-x}[Kcos(x)+Msin(x)]\!$$

Since this does not correspond to our homogeneous solution we can use the '''Basic Rule (a), K 2011, pg. 81''' to solve for the particular solution

$$y_p''+4y_p'+4y_p=r(x)\!$$

where

$$y_p'=e^{-x}[-Ksin(x)+Mcos(x)]-e^{-x}[Kcos(x)+Msin(x)]\!$$

$$y_p'=e^{-x}[-Ksin(x)+Mcos(x)-Kcos(x)-Msin(x)]\!$$

and

$$y_p''=e^{-x}[-Kcos(x)-Msin(x)+Ksin(x)-Mcos(x)]-e^{-x}[Ksin(x)+Mcos(x)-Kcos(x)-Msin(x)]\!$$

$$y_p''=e^{-x}[-Kcos(x)-Msin(x)+Ksin(x)-Mcos(x)+Ksin(x)-Mcos(x)+Kcos(x)+Msin(x)]\!$$

$$y_p''=e^{-x}[2Ksin(x)-2Mcos(x)]\!$$

Plugging these equations back into the differential equation

$$y_p''+4y_p'+4y_p=r(x)\!$$

$$e^{-x}[2Ksin(x)-2Mcos(x)]+4e^{-x}[-Ksin(x)+Mcos(x)-Kcos(x)-Msin(x)]+4e^{-x}[Kcos(x)+Msin(x)]=e^{-x}cos(x)\!$$

$$e^{-x}[2Ksin(x)-2Mcos(x)-4Ksin(x)+4Mcos(x)-4Kcos(x)-4Msin(x)+4Kcos(x)+4Msin(x)]=e^{-x}cos(x)\!$$

$$2Mcos(x)-2Ksin(x)=cos(x)\!$$

from the above equation it is obvious that $$K=0\!$$ and $$M=\frac{1}{2}\!$$

therefore the particular solution to the differential equation is


 * $$y_p(x)=\frac{1}{2}e^{-x}sin(x)\!$$

General Solution
The general solution will be the summation of the homogeneous and particular solutions

$$y(x)=y_h(x)+y_p(x)\!$$

$$y(x)=c_1e^{-2x}+c_2xe^{-2x}+\frac{1}{2}e^{-x}sin(x)\!$$

$$y(x)=e^{-2x}(c_1+c_2x)+\frac{1}{2}e^{-x}sin(x)\!$$

The coefficients $$c_1\!$$ and $$c_2\!$$ can be readily solved for given either initial conditions or boundary value conditions.

Problem Statement
Find a (real) general solution. State which rule you are using. Show each step of your work.
 * $$y''+y'+(\pi ^2+\frac{1}{4})y=e^{-\frac{x}{2}}sin(\pi x)\!$$

Homogeneous Solution
To find the homogeneous solution, $$y_h\!$$, we must find the roots of the equation

$$\lambda^2+\lambda+(\pi ^2+\frac{1}{4})=0\!$$

$$\lambda = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\!$$ with $$a=1, b=1,  c=(\pi ^2+\frac{1}{4})\!$$

$$\lambda = \frac{-1 \pm \sqrt{1^2-4*1*(\pi ^2+\frac{1}{4})}}{2*1}\!$$

$$\lambda =-\alpha \pm i\omega=-\frac{1}{2} \pm \pi i\!$$

We know the homogeneous solution for the case of a double root to be

$$y_h=e^{-\alpha x}[Acos(\omega x)+Bsin(\omega x)]\!$$


 * $$y_h=e^{-\frac{x}{2}}[Acos(\pi x)+Bsin(\pi x)]\!$$

Particular Solution
We have the following excitation
 * $$r(x)=e^{-\frac{x}{2}}sin(\pi x)\!$$

From table 2.1, K 2011, pg. 82, we have
 * $$y_p(x)=e^{-\frac{x}{2}}[Kcos(\pi x)+Msin(\pi x)]\!$$

Since this corresponds to our homogeneous solution we must use the '''Modification Rule (b), K 2011, pg. 81''' to solve for the particular solution

so $$y_p(x)=xe^{-\frac{x}{2}}[Kcos(\pi x)+Msin(\pi x)]\!$$

differentiating

$$y_p'=\pi xe^{-\frac{x}{2}}[-Ksin(\pi x)+Mcos(\pi x)]+(e^{\frac{x}{2}}-\frac{1}{2}xe^{-\frac{x}{2}})[Kcos(\pi x)+Msin(\pi x)]\!$$

$$y_p'=e^{-\frac{x}{2}}(\pi x[-Ksin(\pi x)+Mcos(\pi x)]+(1-\frac{1}{2}x)[Kcos(\pi x)+Msin(\pi x)])\!$$

$$y_p'=e^{-\frac{x}{2}}[-\pi xKsin(\pi x)+\pi xMcos(\pi x)+Kcos(\pi x)+Msin(\pi x)-\frac{1}{2}xKcos(\pi x)-\frac{1}{2}xMsin(\pi x)]\!$$

and

$$y_p''=e^{-\frac{x}{2}}[-\pi ^2xKcos(\pi x)-\pi Ksin(\pi x)-\pi ^2xMsin(\pi x)+\pi Mcos(\pi x)-\pi Ksin(\pi x)+\pi Mcos(\pi x)+\frac{1}{2}\pi xKsin(\pi x)-\frac{1}{2}Kcos(\pi x)-\frac{1}{2}\pi xMcos(\pi x)-\frac{1}{2}Msin(\pi x)]-\frac{1}{2}e^{-\frac{x}{2}}[-\pi xKsin(\pi x)+\pi xMcos(\pi x)+Kcos(\pi x)+Msin(\pi x)-\frac{1}{2}xKcos(\pi x)-\frac{1}{2}xMsin(\pi x)]\!$$

$$y_p''=e^{-\frac{x}{2}}[-\pi ^2xKcos(\pi x)-\pi Ksin(\pi x)-\pi ^2xMsin(\pi x)+\pi Mcos(\pi x)-\pi Ksin(\pi x)+\pi Mcos(\pi x)+\frac{1}{2}\pi xKsin(\pi x)-\frac{1}{2}Kcos(\pi x)-\frac{1}{2}\pi xMcos(\pi x)-\frac{1}{2}Msin(\pi x)+\frac{1}{2}\pi xKsin(\pi x)-\frac{1}{2}\pi xMcos(\pi x)-\frac{1}{2}Kcos(\pi x)-\frac{1}{2}Msin(\pi x)+\frac{1}{4}xKcos(\pi x)+\frac{1}{4}xMsin(\pi x)]\!$$

grouping cosine and sine terms we get

$$y_p''=e^{-\frac{x}{2}}[(2\pi M- \pi xM+(\frac{1}{4}-\pi ^2)xK-K)cos(\pi x)+((\frac{1}{4}-\pi ^2)xM-M+\pi xK-2\pi K)sin(\pi x)]\!$$

and

$$y_p'=e^{-\frac{x}{2}}[(\pi xM+K-\frac{1}{2}xK)cos(\pi x)+(-\pi xK +M-\frac{1}{2}xM)sin(\pi x)]\!$$

next we substitute the above equations into the ODE

$$y_p''+y_p'+(\pi ^2+\frac{1}{4})y_p=r(x)\!$$

$$e^{-\frac{x}{2}}[(2\pi M- \pi xM+(\frac{1}{4}-\pi ^2)xK-K)cos(\pi x)+((\frac{1}{4}-\pi ^2)xM-M+\pi xK-2\pi K)sin(\pi x)]+e^{-\frac{x}{2}}[(\pi xM+K-\frac{1}{2}xK)cos(\pi x)+(-\pi xK +M-\frac{1}{2}xM)sin(\pi x)]+(\pi ^2+\frac{1}{4})xe^{-\frac{x}{2}}[Kcos(\pi x)+Msin(\pi x)]=e^{-\frac{x}{2}}sin(\pi x)\!$$

$$e^{-\frac{x}{2}}[(2\pi M- \pi xM+(\frac{1}{4}-\pi ^2)xK-K)cos(\pi x)+((\frac{1}{4}-\pi ^2)xM-M+\pi xK-2\pi K)sin(\pi x)+(\pi xM+K-\frac{1}{2}xK)cos(\pi x)+(-\pi xK +M-\frac{1}{2}xM)sin(\pi x)+(\pi ^2+\frac{1}{4})x(Kcos(\pi x)+Msin(\pi x))]=e^{-\frac{x}{2}}sin(\pi x)\!$$

$$e^{-\frac{x}{2}}[(2\pi M- \pi xM+(\frac{1}{4}-\pi ^2)xK-K+\pi xM+K-\frac{1}{2}xK+(\pi ^2+\frac{1}{4})xK)cos(\pi x)+((\frac{1}{4}-\pi ^2)xM-M+\pi xK-2\pi K-\pi xK +M-\frac{1}{2}xM+(\pi ^2+\frac{1}{4})xM)sin(\pi x)]=e^{-\frac{x}{2}}sin(\pi x)\!$$

$$2\pi Mcos(\pi x)-2\pi Ksin(\pi x)=sin(\pi x)\!$$

after cancelling terms; we can equate cosine and sine coefficients to get two equations

$$2\pi M=0\!$$

$$-2\pi K=1\!$$

so $$M=0\!$$ and $$K=-\frac{1}{2\pi}\!$$

and the particular solution to the ODE is


 * $$y_p(x)=-\frac{1}{2\pi}xe^{-\frac{x}{2}}cos(\pi x)\!$$

General Solution
The general solution will be the summation of the homogeneous and particular solutions

$$y=y_h+y_p\!$$

$$y=e^{-\frac{x}{2}}[Acos(\pi x)+Bsin(\pi x)]-\frac{1}{2\pi}xe^{-\frac{x}{2}}cos(\pi x)\!$$

$$y=e^{-\frac{x}{2}}[Acos(\pi x)+Bsin(\pi x)-\frac{1}{2\pi}xcos(\pi x)]\!$$

Egm4313.s12.team11.imponenti 04:28, 21 February 2012 (UTC)

Problem 3.9
Solved by Gonzalo Perez

Problem Statement
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.

(K 2011 pg.85 #13)
$$ 8y'' - 6y' + y = 6 \cosh x \! $$              (1)

Initial conditions are:

$$ y(0) = 0.2, y'(0) = 0.05 \! $$

Solution
The general solution of the homogeneous ordinary differential equation is

$$ 8y'' - 6y' + y = 0 \! $$

We can use this information to determine the characteristic equation:

$$ 8 \lambda^2 - 6 \lambda + 1 = 0 \! $$

And proceeding to find the roots,

$$ 4\lambda(2 \lambda - 1) - 1(2 \lambda - 1) = 0 \! $$

Thus, $$ (4 \lambda - 1)(2 \lambda - 1) = 0 \! $$.

Solving for the roots, we find that $$ \lambda = \frac{1}{4}, \frac{1}{2}, \! $$

where the general solution is

$$ y_k = c_1 e^{\frac{1}{4}x} + c_2 e^{\frac{1}{2}x} \! $$.

The solution of $$ y_p \! $$ of the non-homogeneous ordinary differential equation is

$$ x = \frac{e^x+e^{-x}}{2} \! $$.

Using the Sum rule as described in Section 2.7, the above function translates into the following:

$$ y_p = y_{p1} + y_{p2} \! $$, where Table 2.1 tells us that:

$$ y_{p1} = Ae^x \! $$ and $$ y_{p2} = Be^x \! $$.

Therefore, $$ y_{p} = Ae^x + Be^{-x} \! $$.

Now, we can substitute the values ($$ y_p, y_p', y_p'' \! $$) into (1) to get:

$$ 8(Ae^x+Be^{-x}) - 6(Ae^x - Be^{-x}) + Ae^x + Be^{-x} = 6(\frac{e^x + e^{-x}}{2}) \! $$

$$ = 3Ae^x + 15Be^{-x} \! $$

$$ = 3(e^x + e^{-x}) \! $$

Now that we have this equation, we can equate coefficients to find that:

$$ 3A = 3 \! $$

$$ \therefore A = 1 \! $$

$$ B = \frac{3}{15} = \frac{1}{5} \ $$

and thus, $$ y_p = e^x + \frac{1}{5}e^{-x} \! $$

We find that the general solution is in fact:

$$ y = y_k + y_p \! $$

$$ y = c_1 e^{\frac{1}{4}}x + c_2 e^{\frac{1}{2}}x + e^x + 3e^{-x} \! $$

whereas the general solution of the given ordinary differential equation is actually:

$$ y = c_1 e^{\frac{1}{4}}x + c_2 e^{\frac{1}{2}}x + e^x + \frac{1}{5} e^{-x} \! $$

Solving for the initial conditions given and first plugging in $$ y(0) = 0.2 \! $$, we get that:

$$ 0.2 = c_1 e^{\frac{1}{4}(0)} + c_2 e^{\frac{1}{2}(0)} + e^0 + 3e^{(0)} \! $$

$$ 0.2 = c_1 e^{(0)} + c_2 e^{(0)} + e^{(0)} + 3e^{(0)} \! $$

$$ 0.2 = c_1 + c_2 + 1 + \frac{1}{5} \! $$

$$ \therefore c_1 + c_2 = - 1 \! $$.             (2)

And now we can determine the first order ODE :

$$ y' = \frac{1}{4}c_1 e^{\frac{1}{4}(x)} + \frac{1}{2}c_2 e^{\frac{1}{2}(x)} + e^{x} - \frac{1}{5}e^{x} \! $$

The second initial condition that was given to us, $$ y'(0) = 0.05 \! $$ can now be plugged in:

$$ 0.05 = \frac{1}{4}c_1 e^{\frac{1}{4}(0)} + \frac{1}{2}c_2 e^{\frac{1}{2}(0)} + e^{(0)} - \frac{1}{5}e^{(0)} \! $$

$$ 0.05 = \frac{1}{4}c_1 e^{(0)} + \frac{1}{2}c_2 e^{(0)} + e^{(0)} - \frac{1}{5}e^{(0)} \! $$

$$ 0.05 = \frac{1}{4}c_1 + \frac{1}{2}c_2 + 1 - \frac{1}{5} \! $$

$$ \frac{1}{4}c_1 + \frac{1}{2}c_2 = -0.75 \! $$

$$ \therefore c_1 + 2 c_2 = -3 \! $$             (3)

Once we solve (2) and (3), we can get the values:

$$ c_1 = 1, c_2 = - 2 \! $$.

And once we substitute these values, we get the following solution for this IVP:

$$ y = e^{\frac{1}{4}x} - 2 e^{\frac{1}{2}x} + e^{x} + \frac{1}{5} e^{-x} \! $$

(K 2011 pg.85 #14)
$$ y'' + 4y' + 4y = e^{-2x} sin2x \! $$              (1)

Initial conditions are:

$$ y(0)=1, y'(0) = -1.5 \! $$

Solution
The general solution of the homogeneous ordinary differential equation is

$$ y'' + 4y' + 4y = 0 \! $$

We can use this information to determine the characteristic equation:

$$ \lambda^2 + 4 \lambda + 4 = 0 \! $$

And proceeding to find the roots,

$$ (\lambda + 2) (\lambda + 2) = 0 \! $$

Solving for the roots, we find that $$ \lambda = -2, -2 \! $$

where the general solution is:

$$ y_k = c_1 e^{-2x} + c_2 e^{-2x} x \! $$, or:

$$ y_k = (c_1 + c_2 x) e^{-2x} \! $$

Now, according to the Modification Rule and Table 2.1 in Section 2.7, we know that we have to multiply by x to get:

$$ y_p = e^{-2x}(K x cos2x + Mx sin2x) \! $$, since the solution of $$ y_k \! $$ is a double root of the characteristic equation.

We can then derive to get $$ y_p' \! $$:

$$ y_p' = -2 e ^{-2x} (K x cos2x + M x sin2x) + e^{-2x} (Kcos2x-2Kxsin2x+Msin2x+2Mxcos2x) \! $$

$$ y_p' = (-2K+2M) e^{-2x} x cos2x + (-2M-2K) e^-2x xsin2x + Ke^{-2x} cos2x + Me^{-2x} sinx \! $$

Deriving once again to solve for $$ y_p'' \! $$, we get the following:

$$ y_p'' = 4e^{-2x} (Kxcos2x + Mxsin2x) - 2e^{-2x} (Kcos2x - 2Kxsin2x + Msin2x + 2Mxcos2x) - 2e^{-2x} (Kcos2x - 2Kxsin2x + Msin2x + 2Mxcos2x) \! $$

$$ y_p'' = 4e^{-2x} (Kxcos2x + Mxsin2x) - 4e^{-2x} (Kcos2x - 2Kxsin2x + Msin2x + 2Mxcos2x) + e^{-2x} (-4Ksin2x - 4Kxcos2x + 4Mcos2x - 4Mxsin2x) \! $$

$$ y_p'' = (-4K + 4M) e^{-2x} cos2x + (-4K - 4M)e^{-2x} sin2x \ $$

Now, we can substitute the values ($$ y_p, y_p', y_p'' \! $$) into (1) to get:

$$ (-4K + 4M) e^{-2x}cos2x+(-4M-4K)e^{-2x}sin2x+4((-2K+2M)e^{-2x}xcos2x+(-2M-2K)e^{-2x}xsin2x+Ke^{-2x}cos2x+Me^{-2x}sinx) + 4(e^{-2x}(Kxcos2x + Mxsin2x)) = e^{2x}sin2x \! $$

$$ \therefore (-3K+4M)e^{-2x}cos2x + (-3M-4K)e^{-2x}sin2x = e^{-2x}sin2x \! $$

Now that we have this equation, we can equate coefficients to find that:

$$ -3K + 4M =0 \! $$ and $$ -4K - 3M = 1 \! $$

and finally discover that:

$$ M = - \frac{3}{25} \! $$ and $$ K = - \frac{4}{25} \! $$.

Plugging in these values in $$ y_p \! $$, we find that:

$$ y_p = e^{-2x}( - \frac{4}{25}xcos2x - \frac{3}{25}xsin2x) \! $$

And finally, we arrive at the general solution of the given ordinary differential equation:

$$ y = y_k + y_p \! $$

$$ y = (c_1 + c_2 x) e^{-2x}+e^{-2x} (-\frac{4}{25}xcos2x - \frac{3}{25}xsin2x) \! $$

Solving for the initial conditions given and first plugging in $$ y(0) = 1 \! $$, we get that:

$$ 1 = (c_1 + c_2 (0)) e^{-2(0)}+e^{-2(0)} (-\frac{4}{25}(0)cos2(0) - \frac{3}{25}(0)sin2(0)) \! $$

$$ 1 = (c_1 + c_2(0))e^0 \! $$

$$ \therefore c_1 = 1 \! $$

The second initial condition that was given to us, $$ y'(0) = -1.5 \! $$ can now be plugged in:

$$ y' = - \frac{1}{5} e^-2x (10c_1 + 10c_2x - 5c_2 + (3-14x)sin2x+(4-2x)cos(2x)) \! $$

$$ -1.5 = -\frac{1}{5}(10c_1 - 5c_2 + 4) \! $$

$$ \therefore c_2 = -3.5 \! $$

And once we substitute these values, we get the following solution for this IVP:

$$ y = (1 - 3.5 x) e^{-2x}+e^{-2x} (-\frac{4}{25}xcos2x - \frac{3}{25}xsin2x) \! $$

Egm4313.s12.team11.perez.gp 20:34, 22 February 2012 (UTC)