= Solution
The <integrating factor> is $e^{-2x}$, so $(e^{-2x}y)'=e^{(\lambda-2)x}$. Integrating gives \b[the general solution]
$$
\boxed{y=Ce^{2x}+\frac{e^{\lambda x}}{\lambda-2}\quad(\lambda\ne2).}
$$
The <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 written
$$
y=\widetilde C e^{2x}+\frac{e^{\lambda x}-e^{2x}}{\lambda-2}.
$$
Holding $\widetilde C$ fixed, the quotient tends to $xe^{2x}$, the derivative of $e^{\lambda x}$ with respect to $\lambda$ at $2$. Thus \b[the resonant family is $y=e^{2x}(\widetilde C+x)$]. Direct differentiation gives $y'-2y=e^{2x}$, 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 $\widetilde C=2e^{-2}-1$, hence
$$
\boxed{y(x)=e^{2x}\left(x-1+2e^{-2}\right).}
$$
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