ODE No. 2

$$ay(x)+c(-e^{bx})+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.0632209 (sec), leaf count = 34

DSolve[-(c*E^(b*x)) + a*y[x] + Derivative[1][y][x] == 0,y[x],x]
 

$$\left\{\{y(x)\to \frac{c{e}^{x(a+b)-ax}}{a+b}+c_1{e}^{-ax}\}\right\}$$ ✓ Maple : cpu = 0.021 (sec), leaf count = 25

dsolve(diff(y(x),x)+a*y(x)-c*exp(b*x) = 0,y(x))
 

$$y\left(x\right)=(\frac{c{\mathrm{e}}^{(a+b)x}}{a+b}+c_1){\mathrm{e}}^{-ax}$$

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}+ay\left(x\right)=c{e}^{bx}\end{array}$$

Integrating factor $\mu =e^{\int adx}=e^{ax}$. Hence (1) becomes

$$\begin{array}{rl}\frac{d}{dx}\left(\mu y\left(x\right)\right)& =\mu c{e}^{bx}\\ \mu y\left(x\right)& =\int \mu c{e}^{bx}dx+C\end{array}$$

Replacing $\mu$ by $e^{ax}$

$$\begin{array}{rl}y\left(x\right)& =c{e}^{-ax}\int e^{(a+b)x}dx+C{e}^{-ax}\\ & =c{e}^{-ax}\frac{e^{(a+b)x}}{a+b}+C{e}^{-ax}\\ & =\frac{c{e}^{(a+b)x-ax}}{a+b}+C{e}^{-ax}\end{array}$$

Can be reduced to

$$y\left(x\right)=c\frac{e^{bx}}{a+b}+C{e}^{-ax}$$