Differential Equations and The - Braun, Martin 7579 · Problem 2.13.17

The key is recognizing that this integrodifferential equation mixes a derivative and an integral—exactly what the Laplace transform was designed to untangle. When you transform both sides, the integral $\int_0^t y(u)\,du$ becomes $\frac{Y(s)}{s}$ (by Theorem 9), turning the whole equation into an algebraic equation in $Y(s)$. Solve for $Y(s)$, then transform back to recover $y(t)$.

The key is recognizing that this integrodifferential equation mixes a derivative and an integral—exactly what the Laplace transform was designed to untangle. When you transform both sides, the integral $\int_0^t y(u)\,du$ becomes $\frac{Y(s)}{s}$ (by Theorem 9), turning the whole equation into an algebraic equation in $Y(s)$. Solve for $Y(s)$, then transform back to recover $y(t)$.