2.90   ODE No. 90

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

$$x{y}^{\prime }(x)+y(x)-x\sin (x)=0$$ ✓ Mathematica : cpu = 0.012451 (sec), leaf count = 24

$$\left\{\{y(x)\to \frac{\sin (x)-x\cos (x)}{x}+\frac{c_1}{x}\}\right\}$$ ✓ Maple : cpu = 0.011 (sec), leaf count = 17

$$\{y\left(x\right)=\frac{\sin \left(x\right)-\cos \left(x\right)x+\mathit{\_}\mathit{C}\mathit{1}}{x}\}$$

Hand solution

$$x{y}^{\prime }+y=x\sin x$$

Linear first order, exact, separable. $y^{\prime }+\frac{y}{x}=\sin x$, integrating factor $\mu =e^{\int \frac{1}{x}dx}=x$, hence$$\begin{array}{rl}d\left(\mu y\right)& =\mu \sin x\\ xy& =\int x\sin xdx+C\end{array}$$

Using integration by parts. $\int udv=uv-\int vdu$. Let $u=x,dv=\sin x$, hence $du=1,v=-\cos x$, therefore$$\begin{array}{rl}\int x\sin xdx& =-x\cos x+\int \cos x\\ & =-x\cos x+\sin x\end{array}$$

Hence$$\begin{array}{rl}xy& =-x\cos x+\sin x+C\\ y& =\frac{\sin x}{x}-\cos x+\frac{C}{x}\end{array}$$

Verification

restart; 
ode:=x*diff(y(x),x)+y(x)=x*sin(x); 
my_sol:=sin(x)/x-cos(x)+_C1/x; 
odetest(y(x)=my_sol,ode); 
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