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report 1

Report 2

= R2.1: Non-Homogeneous L2-ODE-CC =

This problem is presented in three parts and will thus be solved in three parts:

Part 1 can be found on lecture slide 3-7

Part 2 can be found on lecture slide 3-7

Part 3 can be found on lecture slide 3-9

Given
Two roots:

$$\lambda_1=-2, \ \lambda_2=-2$$

Initial Conditions:

$$y(0)=1, \ y'(0)=0$$

Find the non-homogeneous L2-ODE-CC in standard form and the solution in terms of the initial conditions and the general excitation $$r(x) \ $$.

Methods
From the characteristic equation:

$$(\lambda-\lambda_1)(\lambda-\lambda_2)=0 \ $$

We Have:

$$(\lambda+2)(\lambda-5)=0 \ $$

Expanding to the standard form:

$$\lambda^{2}-3\lambda-10 \ $$

Thus, the non-homogeneous L2-ODE-CC is:

$$y''-3y'-10y=r(x) \ $$

Its corresponding homogeneous solution is:

$$y(x)=c_1e^{5x}+c_2e^{-2x} \ $$

The overall solution $$y(x) \ $$is:

$$y(x)=c_1e^{5x}+c_2e^{-2x}+y_p(x) \ $$

and its derivative $$y'(x) \ $$ is:

$$y'(x)=5c_1e^{5x}-2c_2e^{-2x}+y'_p(x) \ $$

Satisfying the initial conditions:

$$y(0)=c_1+c_2+y_p(0)=1 \ $$            (eq1)

$$y'(0)=5c_1-2c_2+y'p(0) \ $$           (eq2)

Multiplying (eq1) by 2 and adding it to (eq2) yields:

$$2=7c_1+0c_2+2y_p(0)+y'_p(0) \ $$

Solving for $$c_1 \ $$:

$$7c_1=2-2y_p(0)-y'_p(0) \ $$

$$c_1=\frac{1}{7}(2-2y_p(0)-y'_p(0)) \ $$

Substituting $$c_1 \ $$ into (eq1) yields:

$$1=\frac{1}{7}(2-2y_p(0)-y'_p(0))+c_2+y_p(0) \ $$

Rearranging to solve for $$c_2 \ $$:

$$7=2-2y_p(0)-y'_p(0)+7c_2+7y_p(0) \ $$

$$7c_2=7-2+2y_p(0)-7y_p(0)+y'_p(0) \ $$

$$c_2=\frac{1}{7}(5-5y_p(0)+y'_p(0)) \ $$

Solution
Plugging $$c_1,c_2 \ $$ into $$y(x) \ $$:

$$y(x)=\frac{1}{7}(2-2y_p(0)-y'_p(0))e^{5x}+\frac{1}{7}(5-5y_p(0)+y'_p(0))e^{-2x}+y_p(x) \ $$

Given
Consider the particular case of the non-homogeneous L2-ODE-CC solved above with no excitation:

$$r(x)=0 \ $$

and plot the solution.

Methods
Since $$r(x)=0 \ $$, thus $$y_p(x)=0 \ $$ as well as its corresponding 1st and 2nd derivatives.

We can thus solve for the real values of $$c_1, c_2 \ $$:

$$c_1=\frac{2}{7}, \ c_2=\frac{5}{7}$$

Using MATLAB to plot $$y(x) \ $$ we use the following code:

Solution
Plugging in $$c_1, c_2 \ $$ to $$y(x) \ $$ we get:

$$y(x)=\frac{2}{7}e^{5x}+\frac{5}{7}e^{-2x}$$

The corresponding plot for $$y(x) \ $$ is:



Given
Part 3 asks us to generate 3 non-standard & non-homogeneous L2-ODE-CC that still admit the two initial values:

$$\lambda_1=-2, \ \lambda_2=5$$

found in figure 3a on page 3-7 as the two roots in the corresponding characteristic equation.

Methods
The standard form can be found from equation 1 on page 3-8 when $$M=1 \ $$. Non-standard forms can be found using any integer $$M\neq0,1 \ $$.

Thus, three non-standard corresponding equations can be found by using the values:

$$M=2,3,5 \ $$

and the characteristic equation:

$$M(\lambda-\lambda_1)(\lambda-\lambda_2)=0 \ $$.

Plugging in these values of $$M \ $$ we get:

$$M=2 \rightarrow 2\lambda^{2}-6\lambda-20 \ $$

$$M=3 \rightarrow 3\lambda^{2}-9\lambda-30 \ $$

$$M=5 \rightarrow 5\lambda^{2}-15\lambda-50 \ $$

Solution
The corresponding non-standard and non-homogeneous L2-ODE-CC equations with matching roots are:

$$M=2 \rightarrow r(x)=2y''-6y'-20y \ $$

$$M=3 \rightarrow r(x)=3y''-9y'-30y \ $$

$$M=5 \rightarrow r(x)=5y''-15y'-50y \ $$

Author
Solved and typed by Egm4313.s10.team9.delp 04:24, 7 February 2012 (UTC)

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