\[ x y'(x)+y(x)-x \sin (x)=0 \] ✓ Mathematica : cpu = 0.0221124 (sec), leaf count = 24
\[\left \{\left \{y(x)\to \frac {\sin (x)-x \cos (x)}{x}+\frac {c_1}{x}\right \}\right \}\] ✓ Maple : cpu = 0.021 (sec), leaf count = 17
\[y \relax (x ) = \frac {\sin \relax (x )-\cos \relax (x ) x +c_{1}}{x}\]
Hand solution
\[ xy^{\prime }+y=x\sin x \]
Linear first order, exact, separable. \(y^{\prime }+\frac {y}{x}=\sin x\), integrating factor \(\mu =e^{\int \frac {1}{x}dx}=x\), hence\begin {align*} d\left (\mu y\right ) & =\mu \sin x\\ xy & =\int x\sin xdx+C \end {align*}
Using integration by parts. \(\int udv=uv-\int vdu\). Let \(u=x,dv=\sin x\), hence \(du=1,v=-\cos x\), therefore\begin {align*} \int x\sin xdx & =-x\cos x+\int \cos x\\ & =-x\cos x+\sin x \end {align*}
Hence\begin {align*} xy & =-x\cos x+\sin x+C\\ y & =\frac {\sin x}{x}-\cos x+\frac {C}{x} \end {align*}
Verification