Find all solution of the differential equation $y^{\prime\prime} + 4y =\delta$

Question

I am trying to solve this problem.

Find all solution of the differential equation $y^{\prime\prime} + 4y =\delta$, where $\delta$ is Dirac delta function.

The the general solution of the corresponding homogeneous equation, which is $y_H = C_1 cos (2t) +C_2 sin(2t)$.

But I am failing to find the particular solution $y_P$. Please help me.

Answer 1

Hint: Rewrite $$\begin{align} \delta(t)~=~& y^{\prime\prime}(t)+4y(t) \cr ~=~& \left(\frac{d}{dt}+2i\right)\left(\frac{d}{dt}-2i\right)y(t) \cr ~=~& e^{-2it}\frac{d}{dt}e^{4it} \frac{d}{dt}e^{-2it}y(t), \end{align}$$ which can be integrated consecutively to reveal a particular solution $$y(t)~=~\frac{1}{4}{\rm sgn}(t)\sin(2t). $$