\[ a y(x)+c \left (-e^{b x}\right )+y'(x)=0 \] ✓ Mathematica : cpu = 0.0632209 (sec), leaf count = 34
DSolve[-(c*E^(b*x)) + a*y[x] + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {c e^{x (a+b)-a x}}{a+b}+c_1 e^{-a x}\right \}\right \}\] ✓ Maple : cpu = 0.021 (sec), leaf count = 25
dsolve(diff(y(x),x)+a*y(x)-c*exp(b*x) = 0,y(x))
\[y \left (x \right ) = \left (\frac {c \,{\mathrm e}^{\left (a +b \right ) x}}{a +b}+c_{1}\right ) {\mathrm e}^{-a x}\]
Hand solution
\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
\[ y\left ( x\right ) =c\frac {e^{bx}}{a+b}+Ce^{-ax}\]