 Solution

The integrating factor is $e^{-2x}$, so $(e^{-2x}y{)}^{\prime }=e^{(\lambda -2)x}$. Integrating gives the general solutionThe particular solution is only defined up to a homogeneous solution. Subtract $e^{2x}/(\lambda -2)$ from it, absorbing that multiple into the arbitrary constant. The same family can then be writtenHolding $\tilde{C}$ fixed, the quotient tends to $x{e}^{2x}$, the derivative of $e^{\lambda x}$ with respect to $\lambda$ at $2$. Thus the resonant family is $y=e^{2x}(\tilde{C}+x)$. Direct differentiation gives $y^{\prime }-2y=e^{2x}$, so it is the full general solution at resonance. This resonant exponential forcing in a first-order equation limit concerns a reparametrized family, not a fixed value of the original divergent constant $C$. The initial condition gives $\tilde{C}=2e^{-2}-1$, hence