Differential equation

Find all the solutions
y''(x) + y(x) = cos(x) + xy'(0) + y(0)

I found $\frac{1}{2}$ x sen(x) as 1 solution.
I dont know if its the only one.

Quote:

Originally Posted by kezman

Find all the solutions
y''(x) + y(x) = cos(x) + xy'(0) + y(0)

I found $\frac{1}{2}$ x sen(x) as 1 solution.
I dont know if its the only one.

You mean you have found $y = \frac{x}{2} \, {\color{red}\sin} x$ as a particular solution to ${\color{red}y''(x) + y(x) = \cos x}$ .

You're expected to know that the solution to $y''(x) + y(x) = \cos x + xy'(0) + y(0)$ is the homogeneous solution plus a 'total' particular solution.

The homogeneous solution is the solution to $y''(x) + y(x) = 0$ and you're expected to be able to solve this.

You have a particular corresponding to the $\cos x$ term on the right hand side of the DE. Now you need a particular solution corresponding to the term $y'(0) x + y(0)$ . I suggest trying one of the form $y = ax + b$ and getting a and b in terms of y'(0) and y(0). Then the 'total' particular solution is $y = \frac{x}{2} \sin x + ax + b$ .

If you're stuck on finding the homogeneous solution or the 'total' particular solution, post what you've done and where you get stuck.

I think it's "sen" instead of "sin" in Spanish (or maybe some other languages).
(I'm telling this because you highlighted that sin, mr fantastic ;p)