2.11   ODE No. 11

1. Problem in Latex
2. Mathematica input
3. Maple input

$$f(x)y(x)-g(x)+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.0188026 (sec), leaf count = 66

$$\left\{\{y(x)\to \exp ({\int }_1^x-f(K[1])dK[1]){\int }_1^x\exp (-{\int }_1^{K[2]}-f(K[1])dK[1])g(K[2])dK[2]+c_1\exp ({\int }_1^x-f(K[1])dK[1])\}\right\}$$ ✓ Maple : cpu = 0.021 (sec), leaf count = 24

$$\{y\left(x\right)=(\int g\left(x\right){\mathrm{e}}^{\int f\left(x\right)\mathrm{d}x}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}){\mathrm{e}}^{\int -f\left(x\right)\mathrm{d}x}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}+y\left(x\right)f\left(x\right)=g\left(x\right)\end{array}$$

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$$\begin{array}{rl}{e}^{\int f\left(x\right)dx}y\left(x\right)& =\int e^{\int f\left(x\right)dx}g\left(x\right)dx+C\\ y\left(x\right)& =e^{-\int f\left(x\right)dx}\int e^{\int f\left(x\right)dx}g\left(x\right)dx+e^{-\int f\left(x\right)dx}C\\ & =(\int e^{\int f\left(x\right)dx}g\left(x\right)dx+C)e^{-\int f\left(x\right)dx}\end{array}$$