2.7   ODE No. 7

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

$$y^{\prime }(x)+y(x)\cos (x)-e^{-\sin (x)}=0$$ ✓ Mathematica : cpu = 0.0324698 (sec), leaf count = 23

$$\left\{\{y(x)\to x{e}^{-\sin (x)}+c_1{e}^{-\sin (x)}\}\right\}$$ ✓ Maple : cpu = 0.006 (sec), leaf count = 13

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

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}+y\left(x\right)\cos \left(x\right)=e^{-\sin \left(x\right)}\end{array}$$

Integrating factor $\mu =e^{\int \cos dx}=e^{\sin x}$. Hence (1) becomes

$$\frac{d}{dx}\left(\mu y\left(x\right)\right)=\mu e^{-\sin \left(x\right)}$$

Replacing $\mu$ by $e^{\sin x}$ and integrating both sides

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