1.2.11 First order special form ID 1 \(y^{\prime }=g\left ( x\right ) e^{a\left ( x\right ) +by}+f\left ( x\right ) \)

ode internal name "first order special form ID 1"

This is special form which did not fit in any of the above ones. Solved by the substitution \(u=e^{-by}\) which converts the ode to a linear first order ode in \(u\left ( x\right ) \) which is solved, then \(y\) is found. \(b\) must not depend on \(x\) for this to work.