\[ x^2 (-\sin (x))+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.0127858 (sec), leaf count = 15
\[\left \{\left \{y(x)\to c_1 x-x \cos (x)\right \}\right \}\]
✓ Maple : cpu = 0.004 (sec), leaf count = 13
\[ \left \{ y \left ( x \right ) =-\cos \left ( x \right ) x+x{\it \_C1} \right \} \]
\[ xy^{\prime }-y=x^{2}\sin x \]
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 {align*} d\left ( \mu y\right ) & =\mu \sin x\\ \frac {1}{x}y & =\int \sin xdx+C\\ y & =x\left ( C-\cos x\right ) \end {align*}
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