Solving Volterra integral equation

Question

I would like to solve $4u(t)+\int_0^t\sin(t-s)u(s)ds=5t, \ t\geqslant 0$.

Any ideas on how to approach this equation?

Answer 1

Hint. Assume $u(\cdot)$ is continuous over $[0,\infty)$. Then one may differentiate the initial equation twice, using the Leibniz integral rule getting

$$ u''(t)+u(t)=\frac{5}4t, \quad t\geqslant 0, $$

which can be classically solved using $u(0)=0$ and $u'(0)=\dfrac54$.

Answer 2

Through Laplace transform the integral equation becomes $$ 4U(s)+\frac{1}{s^2+1}U(s)=\frac{5}{s^2} $$ that is $$ U(s)=\frac{5}{s^2}\frac{s^2+1}{4s^2+5}=\frac{1}{s^2}+\frac{1}{4s^2+5} $$ and then $$ u(t)=t+\frac{1}{2\sqrt{5}}\sin\left(\frac{\sqrt{5}}{2}\;t\right)\qquad t\ge 0 $$