1. **State the problem:** We need to solve the integro-differential equation $$\dot{y}(t) + \int_0^t y(s) \cos(t - s) \, ds = \cos(t), \quad y(0) = 0.$$ 2. **Recognize the structure:** This is a Volterra integro-differential equation with convolution kernel $\cos(t-s)$. The integral term is a convolution: $$\int_0^t y(s) \cos(t - s) \, ds = (y * \cos)(t).$$ 3. **Use Laplace transform:** Taking Laplace transform $\mathcal{L}\{f(t)\} = F(p)$, and using the convolution theorem: $$\mathcal{L}\{\dot{y}(t)\} = pY(p) - y(0) = pY(p),$$ $$\mathcal{L}\left\{\int_0^t y(s) \cos(t-s) ds\right\} = Y(p) \cdot \frac{p}{p^2 + 1}$$ (since $\mathcal{L}\{\cos t\} = \frac{p}{p^2 + 1}$). 4. **Transform the equation:** $$pY(p) + Y(p) \cdot \frac{p}{p^2 + 1} = \frac{p}{p^2 + 1}.$$ 5. **Factor out $Y(p)$:** $$Y(p) \left(p + \frac{p}{p^2 + 1}\right) = \frac{p}{p^2 + 1}.$$ 6. **Simplify the factor:** $$p + \frac{p}{p^2 + 1} = p \left(1 + \frac{1}{p^2 + 1}\right) = p \frac{p^2 + 1 + 1}{p^2 + 1} = p \frac{p^2 + 2}{p^2 + 1}.$$ 7. **Solve for $Y(p)$:** $$Y(p) = \frac{p}{p^2 + 1} \cdot \frac{p^2 + 1}{p(p^2 + 2)} = \frac{1}{p^2 + 2}.$$ 8. **Inverse Laplace transform:** Recall that $$\mathcal{L}^{-1}\left\{\frac{1}{p^2 + a^2}\right\} = \frac{\sin(at)}{a}.$$ Here, $a = \sqrt{2}$, so $$y(t) = \frac{\sin(\sqrt{2} t)}{\sqrt{2}}.$$ 9. **Check initial condition:** $$y(0) = \frac{\sin(0)}{\sqrt{2}} = 0,$$ which matches the given initial condition. **Final answer:** $$\boxed{y(t) = \frac{\sin(\sqrt{2} t)}{\sqrt{2}}}.$$