Find all the solutions
y''(x) + y(x) = cos(x) + xy'(0) + y(0)
I found x sen(x) as 1 solution.
I dont know if its the only one.
You mean you have found as a particular solution to .
You're expected to know that the solution to is the homogeneous solution plus a 'total' particular solution.
The homogeneous solution is the solution to and you're expected to be able to solve this.
You have a particular corresponding to the term on the right hand side of the DE. Now you need a particular solution corresponding to the term . I suggest trying one of the form and getting a and b in terms of y'(0) and y(0). Then the 'total' particular solution is .
If you're stuck on finding the homogeneous solution or the 'total' particular solution, post what you've done and where you get stuck.