2.9   ODE No. 9

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

$$y^{\prime }(x)-y(x)(a+\sin (\log (x))+\cos (\log (x)))=0$$ ✓ Mathematica : cpu = 0.0187447 (sec), leaf count = 17

$$\left\{\{y(x)\to c_1{e}^{x(a+\sin (\log (x)))}\}\right\}$$

✓ Maple : cpu = 0.04 (sec), leaf count = 14

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

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}-y\left(x\right)[a+\sin (\log \left(x\right))+\cos (\log \left(x\right))]=0\end{array}$$

Integrating factor $\mu =e^{-\int a-\sin (\log \left(x\right))-\cos (\log \left(x\right))dx}=e^{-ax}{e}^{-\int \sin (\log \left(x\right))+\cos (\log \left(x\right))dx}$. To integrate $\int \sin (\log \left(x\right))+\cos (\log \left(x\right))dx$, let $r=\log \left(x\right)$, $\frac{dr}{dx}=\frac{1}{x}$, then $dx=xdr$, But $x=e^r$, hence the integral becomes

$$\begin{array}{rl}\int \sin (\log \left(x\right))+\cos (\log \left(x\right))dx& =\int [\sin \left(r\right)+\cos \left(r\right)]e^{r}dr\\ \text{(2)}& & =\int e^r\sin \left(r\right)dr+\int e^r\cos \left(r\right)dr\end{array}$$

Integrating by parts $\int e^r\cos \left(r\right)dr,$ $\int udv=uv-\int vdu$, Let $u=e^r\to du=e^r$ and $dv=\cos \left(r\right)\to v=\sin \left(r\right)$, hence (2) becomes

$$\begin{array}{rl}\int e^r\sin \left(r\right)dr+\int e^r\cos \left(r\right)dr& =\int e^r\sin \left(r\right)dr+e^r\sin \left(r\right)-\int \sin \left(r\right)e^{r}dr\\ & =e^r\sin \left(r\right)\end{array}$$

Therefore, substituting back $r=\log \left(x\right)$ gives

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

Hence the integration factor is

$$\begin{array}{rl}\mu & =e^{-ax}{e}^{-\int \sin (\log \left(x\right))+\cos (\log \left(x\right))dx}\\ & =e^{-ax}{e}^{-x\sin (\log \left(x\right))}\end{array}$$

Therefore (1) becomes

$$\frac{d}{dx}\left(\mu y\left(x\right)\right)=0$$

Integrating

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