2.92   ODE No. 92

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

$$x^2(-\sin (x))+x{y}^{\prime }(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.0237564 (sec), leaf count = 15

$$\{\{y(x)\to -x\cos (x)+c_{1}x\}\}$$ ✓ Maple : cpu = 0.006 (sec), leaf count = 12

$$\{y\left(x\right)=(c_1-\cos \left(x\right))x\}$$

Hand solution

$$x{y}^{\prime }-y=x^2\sin x$$

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

Verification

restart; 
ode:=x*diff(y(x),x)-y(x)=x^2*sin(x); 
my_sol:=x*(_C1-cos(x)); 
odetest(y(x)=my_sol,ode); 
0