\[ a y(x)+c \left (-e^{b x}\right )+y'(x)=0 \] ✓ Mathematica : cpu = 0.0323651 (sec), leaf count = 33
\[\left \{\left \{y(x)\to \frac {e^{-a x} \left (c e^{x (a+b)}+c_1 (a+b)\right )}{a+b}\right \}\right \}\]
✓ Maple : cpu = 0.163 (sec), leaf count = 25
\[ \left \{ y \left ( x \right ) = \left ( {\frac {c{{\rm e}^{ \left ( a+b \right ) x}}}{a+b}}+{\it \_C1} \right ) {{\rm e}^{-ax}} \right \} \]
\begin {equation} \frac {dy}{dx}+ay\left ( x\right ) =ce^{bx}\tag {1} \end {equation}
Integrating factor \(\mu =e^{\int adx}=e^{ax}\). Hence (1) becomes
\begin {align*} \frac {d}{dx}\left ( \mu y\left ( x\right ) \right ) & =\mu ce^{bx}\\ \mu y\left ( x\right ) & =\int \mu ce^{bx}dx+C \end {align*}
Replacing \(\mu \) by \(e^{ax}\)
\begin {align*} y\left ( x\right ) & =ce^{-ax}\int e^{\left ( a+b\right ) x}dx+Ce^{-ax}\\ & =ce^{-ax}\frac {e^{\left ( a+b\right ) x}}{a+b}+Ce^{-ax}\\ & =\frac {ce^{\left ( a+b\right ) x-ax}}{a+b}+Ce^{-ax} \end {align*}
Can be reduced to
\[ y\left ( x\right ) =c\frac {e^{bx}}{a+b}+Ce^{-ax}\]