Find the general solution of the equation

$$\frac{d y}{d x}-2 y=e^{\lambda x}$$

where $\lambda$ is a constant not equal to 2 .

By subtracting from the particular integral an appropriate multiple of the complementary function, obtain the limit as $\lambda \rightarrow 2$ of the general solution of $(*)$ and confirm that it yields the general solution for $\lambda=2$.

Solve equation $(*)$ with $\lambda=2$ and $y(1)=2$.