Integro-Differential Equations

Question

I was attempting to solve the following integro-differential equation using convolutions. My answer also had a convolution which did not seem right and was wondering if someone would check my process.

Problem with initial work

My final solution

Answer 1

$$y'(t)=1-\int_0^t y(t-\tau ) \exp (-2 \tau ) \, d\tau$$ Laplace Transform: $$s \left(\mathcal{L}_t[y(t)](s)\right)-y(0)=\frac{1}{s}-\frac{\mathcal{L}_t[y(t)](s)}{2+s}$$

We have $y(0)=1$ and solve for $\mathcal{L}_t[y(t)](s)$:

$$\mathcal{L}_t[y(t)](s)=\frac{2+s}{s (1+s)}$$ $$\mathcal{L}_t[y(t)](s)=\frac{2}{s}-\frac{1}{1+s}$$

Inverse Laplace Transform:

$$y(t)=2-e^{-t}$$