2.92   ODE No. 92

\[ x^2 (-\sin (x))+x y'(x)-y(x)=0 \]

✓ Mathematica : cpu = 0.0171002 (sec), leaf count = 15

DSolve[-(x^2*Sin[x]) - y[x] + x*Derivative[1][y][x] == 0,y[x],x]
 

\[\{\{y(x)\to -x \cos (x)+c_1 x\}\}\]

✓ Maple : cpu = 0.007 (sec), leaf count = 12

dsolve(x*diff(y(x),x)-y(x)-x^2*sin(x) = 0,y(x))
 

\[y \left (x \right ) = \left (-\cos \left (x \right )+c_{1} \right ) x\]

Hand solution

\[ xy^{\prime }-y=x^{2}\sin x \]

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); 
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