Theoretical question regarding unbounded solutions to a differential equation I am lost regarding the following question:
Let $a$ and $b$ be real numbers. It is known that the function $y''+ay'+by=\sin(x)$ has a solution that is a non-bounded function. It is also known that the difference between any two solutions is a bounded function all across $\mathbb{R}$. Find $a$ and $b$. Carefully explain each one of your considerations.
I understand that this has to do with $a^2-4b>0$ so that the homogenous solution comes out non-bounded. Also, $a$ should not be equal to $0$, for the formula computing the roots to come out defined. Beyond that point (and particularly regarding the "difference between any two solutions"), I do not understand where I am supposed to take it to… Many thanks!

The characteristic equation $$\lambda ^2 +a\lambda +b =0$$ has to have pure imaginary solutions for the homogeneous solution to be bounded and the general solution to be unbounded.
Thus you have to have $$a=0$$ and $b=1$
The equation $$y''+y=\sin x$$ has its general solution as $$ y=C_1 \sin x + C_2 \cos x -(1/2)x\cos x$$
