User:EGM6321.f10.team6.cook/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. $$\begin{align}\scriptstyle 1-15+85-225+274-120=0\end{align}$$.