# Thread: Second order differential equation

1. ## Second order differential equation

Hi everybody, I have absolutely no idea how to tackle this problem:

Let p(x) be a solution of the differential equation $y\prime\prime + yy\prime = x^{3}$ with initial conditions $y(-1) = 1$ and $y\prime(-1)=2$. Find $p\prime\prime(-1)$ and $p\prime\prime\prime(-1)$.

Any help would be greatly appreciated.
Thank you!

2. Originally Posted by gundu24
Hi everybody, I have absolutely no idea how to tackle this problem:

Let p(x) be a solution of the differential equation $y\prime\prime + yy\prime = x^{3}$ with initial conditions $y(-1) = 1$ and $y\prime(-1)=2$. Find $p\prime\prime(-1)$ and $p\prime\prime\prime(-1)$.

Any help would be greatly appreciated.
Thank you!
Did you try anything on this?

Since p(x) satisfies the equation, $p''(x)= x^3- p(x)p'(x)$. You know the values of x, y, and y' at -1 so you can directly calculate the right side.

To find p'"(-1), Differentiate both sides of the equation above and evaluate at x=-1.

3. The problem is quite interesting because, applying the procedure described by HalsofIvy, You can iteratively compute the quantities $p^{(n)} (-1)$ and that allows You to obtain the Taylor expansion of $p(x)$ around $x=-1$ ...

$p(x)= \sum_{n=0}^{\infty} \frac{p^{(n)} (-1)}{n!}\cdot (x+1)^{n}$ (1)

Kind regards

$\chi$ $\sigma$

4. Yes, that's a standard, though very slow and not very practical, method of finding power series solutions to differential equations.