Bernoulli equation with exponential forcing ($y^{\prime }=ay+q{e}^{\lambda x}{y}^p$)

For $p\ne 1$ on a positive real branch, put $v=y^{1-p}$ and $\kappa =(1-p)a$. The Bernoulli power substitution gives $v^{\prime }-\kappa v=(1-p)q{e}^{\lambda x}$. Its integrating factor is $e^{-\kappa x}$, yieldingThe resonant case needs an extra factor x because the forcing has the same exponential dependence as the homogeneous solution. Recover $y=v^{1/(1-p)}$ only on an appropriate real branch. The case $p=1$ is already a first-order linear differential equation; a zero solution or zero-crossing behavior must be checked in the original equation separately.