2.6.10.6 ✓ Maple. Time used: 0.001 (sec). Leaf size: 21

ode:=1/x*diff(y(x),x) = y(x)*sin(x^2-1)-2*y(x)/x^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = c_1 \,{\mathrm e}^{-\frac {\cos \left (x^{2}-1\right )}{2}-\frac {4 x^{{3}/{2}}}{3}} \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d x}y \left (x \right )}{x}=y \left (x \right ) \sin \left (x^{2}-1\right )-\frac {2 y \left (x \right )}{\sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\left (y \left (x \right ) \sin \left (x^{2}-1\right )-\frac {2 y \left (x \right )}{\sqrt {x}}\right ) x \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d x}y \left (x \right )}{y \left (x \right )}=\left (\sqrt {x}\, \sin \left (\left (x -1\right ) \left (x +1\right )\right )-2\right ) \sqrt {x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\frac {d}{d x}y \left (x \right )}{y \left (x \right )}d x =\int \left (\sqrt {x}\, \sin \left (\left (x -1\right ) \left (x +1\right )\right )-2\right ) \sqrt {x}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y \left (x \right )\right )=-\frac {\cos \left (\left (x -1\right ) \left (x +1\right )\right )}{2}-\frac {4 x^{{3}/{2}}}{3}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )={\mathrm e}^{-\frac {\cos \left (\left (x -1\right ) \left (x +1\right )\right )}{2}-\frac {4 x^{{3}/{2}}}{3}+\mathit {C1}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )={\mathrm e}^{-\frac {\cos \left (x^{2}-1\right )}{2}-\frac {4 x^{{3}/{2}}}{3}+\mathit {C1}} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-\frac {\cos \left (x^{2}-1\right )}{2}-\frac {4 x^{{3}/{2}}}{3}} \end {array} \]