#### 2.11   ODE No. 11

$f(x) y(x)-g(x)+y'(x)=0$ Mathematica : cpu = 0.0191528 (sec), leaf count = 66

$\left \{\left \{y(x)\to c_1 \exp \left (\int _1^x-f(K[1])dK[1]\right )+\exp \left (\int _1^x-f(K[1])dK[1]\right ) \int _1^x\exp \left (-\int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]\right \}\right \}$ Maple : cpu = 0.018 (sec), leaf count = 24

$\left \{ y \left ( x \right ) = \left ( \int \!g \left ( x \right ) {{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{\it \_C1} \right ) {{\rm e}^{\int \!-f \left ( x \right ) \,{\rm d}x}} \right \}$

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*}