2.3   ODE No. 3

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

$$ay(x)-b\sin (cx)+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.0497953 (sec), leaf count = 40

$$\left\{\{y(x)\to \frac{b(a\sin (cx)-c\cos (cx))}{a^2+c^2}+c_1{e}^{-ax}\}\right\}$$

✓ Maple : cpu = 0.211 (sec), leaf count = 37

$$\{y\left(x\right)={\mathrm{e}}^{-ax}\mathit{\_}\mathit{C}\mathit{1}+\frac{b(\sin \left(cx\right)a-c\cos \left(cx\right))}{a^2+c^2}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}+ay\left(x\right)=b\sin \left(cx\right)\end{array}$$

Integrating factor $\mu =e^{\int adx}=e^{ax}$. Hence (1) becomes

$$\begin{array}{rl}\frac{d}{dx}\left(\mu y\left(x\right)\right)& =\mu b\sin \left(cx\right)\\ \mu y\left(x\right)& =b\int \mu \sin \left(cx\right)dx+C\end{array}$$

Replacing $\mu$ by $e^{ax}$

$$\begin{array}{}\text{(2)}& y\left(x\right)=b{e}^{-ax}\int e^{ax}\sin \left(cx\right)dx+C{e}^{-ax}\end{array}$$

Using $\sin \left(cx\right)=\frac{e^{icx}-e^{-icx}}{2i}$ then $$\begin{array}{rl}\int e^{ax}\sin \left(cx\right)dx& =\int \frac{e^{(ic+a)x}-e^{(-ic+a)x}}{2i}dx\\ & =\frac{1}{2i}(\frac{e^{(ic+a)x}}{ic+a}-\frac{e^{(-ic+a)x}}{-ic+a})\\ & =\frac{1}{2i}{e}^{ax}(\frac{e^{icx}}{ic+a}-\frac{e^{-icx}}{-ic+a})\\ & =\frac{1}{2i}{e}^{ax}\left(\frac{e^{icx}(-ic+a)-e^{-icx}(ic+a)}{(ic+a)(-ic+a)}\right)\\ & =\frac{1}{2i}{e}^{ax}\left(\frac{-ic{e}^{icx}+a{e}^{icx}-ic{e}^{-icx}-a{e}^{-icx}}{(c^2+a^2)}\right)\\ & =\frac{1}{2i}{e}^{ax}\left(\frac{-ic(e^{icx}+e^{-icx})+a(e^{icx}-e^{-icx})}{(c^2+a^2)}\right)\\ & =\frac{e^{ax}}{(c^2+a^2)}(\frac{-ic(e^{icx}+e^{-icx})}{2i}+\frac{a(e^{icx}-e^{-icx})}{2i})\\ & =\frac{e^{ax}}{(c^2+a^2)}(-c\cos cx+a\sin cx)\end{array}$$

Therefore (2) becomes

$$\begin{array}{rl}y\left(x\right)& =b{e}^{-ax}\left[\frac{e^{ax}}{(c^2+a^2)}(-c\cos cx+a\sin cx)\right]+C{e}^{-ax}\\ & =\frac{b}{(c^2+a^2)}(-c\cos cx+a\sin cx)+C{e}^{-ax}\end{array}$$