\[ f(x) y(x)-g(x)+y'(x)=0 \] ✓ Mathematica : cpu = 0.0288013 (sec), leaf count = 66
DSolve[-g[x] + f[x]*y[x] + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \exp \left (\int _1^x-f(K[1])dK[1]\right ) \int _1^x\exp \left (-\int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]+c_1 \exp \left (\int _1^x-f(K[1])dK[1]\right )\right \}\right \}\] ✓ Maple : cpu = 0.028 (sec), leaf count = 24
dsolve(diff(y(x),x)+f(x)*y(x)-g(x) = 0,y(x))
\[y \left (x \right ) = \left (\int g \left (x \right ) {\mathrm e}^{\int f \left (x \right )d x}d x +c_{1}\right ) {\mathrm e}^{\int -f \left (x \right )d x}\]
Hand solution
\begin {equation} \frac {dy}{dx}+y\left ( x\right ) f\left ( x\right ) =g\left ( x\right ) \tag {1} \end {equation}
Integrating factor \(\mu =e^{\int f\left ( x\right ) dx}\). Therefore (1) becomes\[ \frac {d}{dx}\left ( e^{\int f\left ( x\right ) dx}y\left ( x\right ) \right ) =e^{\int f\left ( x\right ) dx}g\left ( x\right ) \] Integrating\begin {align*} e^{\int f\left ( x\right ) dx}y\left ( x\right ) & =\int e^{\int f\left ( x\right ) dx}g\left ( x\right ) dx+C\\ y\left ( x\right ) & =e^{-\int f\left ( x\right ) dx}\int e^{\int f\left ( x\right ) dx}g\left ( x\right ) dx+e^{-\int f\left ( x\right ) dx}C\\ & =\left ( \int e^{\int f\left ( x\right ) dx}g\left ( x\right ) dx+C\right ) e^{-\int f\left ( x\right ) dx} \end {align*}