Solve $\int_0^xQ(x-y)[Q(y)-2\sin ay]dy=x\cos ax$ using Laplace Transform

Question

$$\int_0^xQ(x-y)[Q(y)-2\sin ay]dy=x\cos ax$$ Taking Laplace Transform of the Integral Equation gives: $$\mathcal{L}\{Q\}[\mathcal{L}\{Q\}-\frac{2a}{p^2+a^2}]=\frac{p^2-a^2}{{p^2+a^2}^2}$$

This was where I got stuck on how to simplify further for $Q$

Please can someone simplify further or show me where i got it wrong

Answer 1

I've been able to simplify further. It's actually simple. I transformed the equation into a quadratic equation:

$$\mathcal{L}\{Q\}[\mathcal{L}\{Q\}-\frac{2a}{p^2+a^2}]=\frac{p^2-a^2}{{p^2+a^2}^2}$$

Which becomes:

$$\mathcal{L}\{Q\}^2-\frac{2a}{p^2+a^2}\mathcal{L}\{Q\}-\frac{p^2-a^2}{{p^2+a^2}^2}=0$$

Solving this quadratic equation gives:

$$\mathcal{L}\{Q\}=\frac{a\pm p}{p^2+a^2} $$

Thus taking the inverse Laplace transformation gives:

$$Q=\sin ax \pm\cos ax$$