2.1.7.3 ✓ Maple. Time used: 0.001 (sec). Leaf size: 24

ode:=diff(v(u),u)+u^2*v(u) = sin(u); 
dsolve(ode,v(u), singsol=all);
 

\begin{align*} v = \left (\int \sin \left (u \right ) {\mathrm e}^{\frac {u^{3}}{3}}d u +c_1 \right ) {\mathrm e}^{-\frac {u^{3}}{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 {d}{d u}v \left (u \right )+u^{2} v \left (u \right )=\sin \left (u \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d u}v \left (u \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )=-u^{2} v \left (u \right )+\sin \left (u \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} v \left (u \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & \frac {d}{d u}v \left (u \right )+u^{2} v \left (u \right )=\sin \left (u \right ) \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (u \right ) \\ {} & {} & \mu \left (u \right ) \left (\frac {d}{d u}v \left (u \right )+u^{2} v \left (u \right )\right )=\mu \left (u \right ) \sin \left (u \right ) \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d u}\left (v \left (u \right ) \mu \left (u \right )\right ) \\ {} & {} & \mu \left (u \right ) \left (\frac {d}{d u}v \left (u \right )+u^{2} v \left (u \right )\right )=\left (\frac {d}{d u}v \left (u \right )\right ) \mu \left (u \right )+v \left (u \right ) \left (\frac {d}{d u}\mu \left (u \right )\right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \frac {d}{d u}\mu \left (u \right ) \\ {} & {} & \frac {d}{d u}\mu \left (u \right )=\mu \left (u \right ) u^{2} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (u \right )={\mathrm e}^{\frac {u^{3}}{3}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} u \\ {} & {} & \int \left (\frac {d}{d u}\left (v \left (u \right ) \mu \left (u \right )\right )\right )d u =\int \mu \left (u \right ) \sin \left (u \right )d u +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & v \left (u \right ) \mu \left (u \right )=\int \mu \left (u \right ) \sin \left (u \right )d u +\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \left (u \right ) \\ {} & {} & v \left (u \right )=\frac {\int \mu \left (u \right ) \sin \left (u \right )d u +\mathit {C1}}{\mu \left (u \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (u \right )={\mathrm e}^{\frac {u^{3}}{3}} \\ {} & {} & v \left (u \right )=\frac {\int {\mathrm e}^{\frac {u^{3}}{3}} \sin \left (u \right )d u +\mathit {C1}}{{\mathrm e}^{\frac {u^{3}}{3}}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & v \left (u \right )={\mathrm e}^{-\frac {u^{3}}{3}} \left (\int {\mathrm e}^{\frac {u^{3}}{3}} \sin \left (u \right )d u +\mathit {C1} \right ) \end {array} \]