2.5   ODE No. 5

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

$$y^{\prime }(x)+y(x)\cos (x)-e^{2x}=0$$ ✓ Mathematica : cpu = 0.382991 (sec), leaf count = 39

$$\left\{\{y(x)\to e^{-\sin (x)}{\int }_1^x{e}^{2K[1]+\sin (K[1])}dK[1]+c_1{e}^{-\sin (x)}\}\right\}$$ ✓ Maple : cpu = 0.112 (sec), leaf count = 21

$$\{y\left(x\right)=(\int {\mathrm{e}}^{2x+\sin \left(x\right)}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}){\mathrm{e}}^{-\sin \left(x\right)}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}+y\left(x\right)\cos \left(x\right)=e^{2x}\end{array}$$

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

$$\frac{d}{dx}\left(e^{\sin \left(x\right)}y\left(x\right)\right)=e^{\sin \left(x\right)}{e}^{2x}$$

Integrating both sides

$$\begin{array}{rl}{e}^{\sin \left(x\right)}y\left(x\right)& =\int e^{\sin \left(x\right)}{e}^{2x}+C\\ y\left(x\right)& =e^{-\sin \left(x\right)}\int e^{2x+\sin \left(x\right)}+C{e}^{-\sin \left(x\right)}\end{array}$$