User:Egm6321.f10.team6.Kim.MK/hw5

= Problem 1 - Find $$\begin{align}y_{xxxxx}\end{align}$$ =

Statement
Find $$\begin{align}y_{xxxxx}\end{align}$$ in terms of derivative of y with respect to t, where $$\begin{align}y(x(t))\end{align}$$ and $$\begin{align} x=\textbf{e}^{t}\end{align}$$.

Solution
Applying the chain rule to $$\begin{align}y(x(t))\end{align}$$

Then solving for $$\begin{align}y_x\end{align}$$

Taking the derivative with respect to $$\begin{align}x\end{align}$$

Then solving for $$\begin{align}y_{xx}\end{align}$$

Taking another derivative with respect to $$\begin{align}x\end{align}$$

And again solving for higher order derivative with respect to $$\begin{align}x\end{align}$$

Repeating the process again for the fourth derivative

And again solving for higher order derivative with respect to $$\begin{align}x\end{align}$$

Repeating the process one final time for the fifth derivative

And solving for fifth order derivative with respect to $$\begin{align}x\end{align}$$

Note that the coefficients of the equation for the derivative with respect to $$\begin{align}x\end{align}$$ of each order form a Diophantine equation which sums to zero. This is expected as if $$\begin{align}y\end{align}$$ is constant with respect to $$\begin{align}t\end{align}$$ it should also be constant with respect to $$\begin{align}x\end{align}$$

General Solution Method
There is a 'general' recursive algorithm for finding the coefficients of the partial derivatives with respect to $$\begin{align}t\end{align}$$. Expressing the nth partial with respect to $$\begin{align}x\end{align}$$ as a sum of partials with respect to $$\begin{align}t\end{align}$$.

The first coefficient is always one

The second term initiates the recursion and is

Then the next coefficient is found by multiplying each term by the terms to the right, dropping the terms where there are no terms to the right. To demonstrate $$\begin{align}y_{xxxxxx}\end{align}$$ ($$\begin{align}y_{n=6}\end{align}$$) will be found without using $$\begin{align}y_{xxxxx}\end{align}$$.

Then each term times the polynomial to its right

Again applying the recursive rule

Applying again for the fourth term

And once again for the last term

Thus $$y_{xxxxxx}$$ is

The solution is checked by noting the sum of coefficients is zero, i.e. $$\begin{align}1-15+85-225+274-120=0\end{align}$$.

Problem Statement
A general spring-mass-damper system has spring constant k, mass m, and damping coefficient c. The equation of motion is given as :

Which may be rewritten as follows :

Where the homogenouse ODE is given as:

Part 1
1. Solve homogeneous ODE given in equation 5.3 by assuming $$y={{e}^{rt}}$$ to calculate homogeneous equation and obtain roots.

2. If only one root is obtained use variation of parameters to obtain both homogenous solutions.

3. Find the particular solution for equation 5.2.

Part 2
1. For equation 5.4 find the values of  $${_{1}}$$ and $${{{\bar{a}}}_{0}}$$ in terms of  $${{a}_{2}}$$, $${{a}_{1}}$$, and $${{a}_{0}}$$.

2. Find quadratic equation for $$\alpha $$.

3. Reduce order of equation.

4. Apply integrating factor method to part 2.3

5. Show that:

Where $$\alpha $$ and $$\beta $$ are roots of the equation for $$\alpha $$ in part 2.2

6. Deduce and expression for the particular solution for a general excitation  $${{f}_{\left( t \right)}}$$

7. Verify the table presented in fall 2009.

8. Solve $${{f}_{\left( t \right)}}={{e}^{\left( -{{t}^{2}} \right)}}$$ such that:

8.1 The roots for equation 5.1 are obtained from $$\left( r+1 \right)\left( r-2 \right)=0$$.

8.2 The roots for equation 5.1 are obtained from $${{\left( r-4 \right)}^{2}}=0$$. .

Part 1.1
We begin by solving the homogeneous equation given in equation 5.3. First we assume a solution to the equation fits the form:

This yields:

We can divide by a common factor of $${{e}^{rt}}$$ to obtain the auxiliary equation:

In order to calculate the roots of equation 5.9 we must apply the quadratic formula in equation 5.10.

This yields:

Which after some simplification results in the following equation for the roots to equation 5.3:

Part 1.2
The number of roots that are obtained from the auxiliary equation for equation 5.3 depend on the value of $$\varsigma $$.

$$\begin{align} & if: \\ & \varsigma =1\text{ }\to \text{ one real root equal to }{{\omega }_{o}} \\ & \varsigma >1\to \text{ two real roots} \\ & \varsigma <1\to \text{ two complex roots} \\ \end{align}$$

Therefore, if $$\varsigma $$ is unknown it is not possible to know the number of roots.

Given
and

Find
To show that the expressions for $$\begin{align}y_{p}(x)\end{align}$$ is equivalent

Solution
Write (6.3) in follows,

According to (6.1),

Differentiation rules shows that

Along with (6.4) we can get

Therefore (6.5) becomes,

Integration by parts for (6.2)

Comparing (6.9) and (6.10) We can see they are equivalent.