Solve Integral Equation With Convolution and a constant added

Question

$7t + 8/5 $$\int_0^t \cos (a(t - \tau)) y(\tau) d\tau$$ = y(t)$ , for $ a>10^{50}$

Im sort of confused by working with the constant $a$.

i used convolution theorem and applied laplace, giving it: $ 7t + 8/5[(s/(s^2 + a^2) Y(s)] = Y(s) $

Answer 1

The convolution property says that $$L\!\left[\int_0^tf(\tau)\,g(t-\tau)\,d\tau\right]=F(s)\,G(s),$$ where \begin{align*} L[f(t)]&=F(s)\\ L[g(t)]&=G(s). \end{align*} So in your case, that would mean, taking the LT of the entire equation: \begin{align*} \frac{7}{s^2}+\frac85\,L[\cos(at)]\,Y(s)&=Y(s)\\ \frac{7}{s^2}+\frac{8\,s\,Y(s)}{s^2+a^2}&=Y(s). \end{align*} Solving for $Y(s)$ yields $$Y(s)=\frac{7(s^2+a^2)}{s^2(s^2-8s+a^2)}.$$ Now can you finish?