# Thread: second order differential equation for IVP

1. ## second order differential equation for IVP

hi guys, Im wondering how to solve this problem

''Consider the following initial value problem:

By observing that the first two terms on the right-hand side of the equation form a total derivative of a function, find the analytical solution of the problem.''

thanks

2. Originally Posted by hazeleyes
hi guys, Im wondering how to solve this problem

''Consider the following initial value problem:

By observing that the first two terms on the right-hand side of the equation form a total derivative of a function, find the analytical solution of the problem.''

thanks
What the hint is tell you is notice that

$\displaystyle \frac{2}{x}y'-\frac{2}{x^2}y=\frac{d}{dx}\left( \frac{2}{x}y\right)$

So using this identity we get that

$\displaystyle \frac{d}{dx}(y')=\frac{d}{dx}\left( \frac{2}{x}y \right)-\frac{1}{x^2}$

Now move everything to the left hand side

$\displaystyle \frac{d}{dx}\left(y'- \frac{2}{x}y \right)=-\frac{1}{x^2}$

Now just integrate to get a first order ode

Can you finish from here?