#### 2.9   ODE No. 9

$y'(x)-y(x) (a+\sin (\log (x))+\cos (\log (x)))=0$ Mathematica : cpu = 0.015432 (sec), leaf count = 19

$\left \{\left \{y(x)\to c_1 e^{a x+x \sin (\log (x))}\right \}\right \}$ Maple : cpu = 0.013 (sec), leaf count = 14

$\left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{x \left ( \sin \left ( \ln \left ( x \right ) \right ) +a \right ) }} \right \}$

Hand solution

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

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

\begin {align} \int \sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) dx & =\int \left [ \sin \left ( r\right ) +\cos \left ( r\right ) \right ] e^{r}dr\nonumber \\ & =\int e^{r}\sin \left ( r\right ) dr+\int e^{r}\cos \left ( r\right ) dr\tag {2} \end {align}

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

\begin {align*} \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 {align*}

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

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

Hence the integration factor is

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

Therefore (1) becomes

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

Integrating

\begin {align*} y\left ( x\right ) e^{-ax}e^{-x\sin \left ( \log \left ( x\right ) \right ) } & =C\\ y\left ( x\right ) & =Ce^{ax}e^{x\sin \left ( \log \left ( x\right ) \right ) }\\ & =Ce^{ax+x\sin \left ( \log \left ( x\right ) \right ) }\\ & =Ce^{x\left ( a+\sin \left ( \log \left ( x\right ) \right ) \right ) } \end {align*}