Find all the solutions
y''(x) + y(x) = cos(x) + xy'(0) + y(0)
I found $\displaystyle \frac{1}{2}$x sen(x) as 1 solution.
I dont know if its the only one.
You mean you have found $\displaystyle y = \frac{x}{2} \, {\color{red}\sin} x $ as a particular solution to $\displaystyle {\color{red}y''(x) + y(x) = \cos x}$.
You're expected to know that the solution to $\displaystyle y''(x) + y(x) = \cos x + xy'(0) + y(0)$ is the homogeneous solution plus a 'total' particular solution.
The homogeneous solution is the solution to $\displaystyle y''(x) + y(x) = 0$ and you're expected to be able to solve this.
You have a particular corresponding to the $\displaystyle \cos x$ term on the right hand side of the DE. Now you need a particular solution corresponding to the term $\displaystyle y'(0) x + y(0)$. I suggest trying one of the form $\displaystyle y = ax + b$ and getting a and b in terms of y'(0) and y(0). Then the 'total' particular solution is $\displaystyle y = \frac{x}{2} \sin x + ax + b$.
If you're stuck on finding the homogeneous solution or the 'total' particular solution, post what you've done and where you get stuck.