I am trying to solve the Volterra integral equation $$f(t)=t^2-\int_{0}^{t} (t-s)f(s) \ ds,$$ using the Laplace transformation.
I first rewrite the equation as $$f(t)=t^2-(f*t)(t),$$ where $*$ denotes the convolution of $f$ and $t$. Taking the Laplace transform of each term (Laplace transform is linear), \begin{align} \mathcal{L}(f)(z)&=\mathcal{L}(t^2)(z)-\mathcal{L}(f*t)(z) \\ \mathcal{L}(f)(z)&=\frac{2}{z^3}-\frac{\mathcal{L}(f)(z)}{z^2} \\ \implies \mathcal{L}(f)(z)&=\frac{2}{z(1+z^2)}. \\ \end{align} Using the inversion formula $$f(t)=\sum_{j=1}^{3}\text{Res}(\mathcal{L}(f)(z)e^{zt},z=a_j),$$ we conclude that $$f(t)=2-(e^{it}+e^{-it})=2-2\cos(t).$$
Is this solution correct? I have minimal experience using the Laplace transform and would like to know of any errors in my logic before I attempt more complicated questions.