# How to solve a differential equation containing convolution?

I got a differential equation as $$C_0+C_1*y(x)+C_2*∫_0^xh(x)\frac{d\tau }{\sqrt{\tau }} =C_3*y''(x)$$ With parameters and boundary conditions as $$C_0=C_1=C_2=C_3=1,y(0)=0,y'(0)=0,y''(0)=1$$ Where the $h(x)$ was $$h(x)=\int y (x-\tau ) \, d\tau$$

One of my friend told me that $h(x)$ was a convolution, and by using $f (x) = 1$ & $g (x) = y (x)$, their convolution could be handled as $$h(x)=(f⨯g)(x)=∫_0^xf(τ)g(x-τ)d\tau$$ $$h(x)=f(x)\times g(x)=g(x)\times f(x)=g(x)\times 1 =g(x)=y(x)$$

Well, I am not familiar with the convolution theory and not sure if his suggestion was right.

Is there any suggestion that I could solve the equation? Solving it either explicitly or numerically would be OKay.

Thank you.

## 1 Answer

Hint.

Given the Laplace transform $L(\cdot)$ we have

$$L\left(\int_{t=0}^{t=x}f(t)g(x-t)dt\right) = L(f)L(g) = F(s)G(s)$$