Internal problem ID [6708]
Book: Second order enumerated odes
Section: section 2
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-y^{\prime }+4 y x^{3}-8 x^{3} \left (\sin ^{2}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 102
dsolve(x*diff(y(x),x$2)-diff(y(x),x)+4*x^3*y(x)=8*x^3*sin(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \sin \left (x^{2}\right ) c_{2}+\cos \left (x^{2}\right ) c_{1}+\frac {\left (-\FresnelC \left (\frac {\sqrt {2}\, \left (x -1\right )}{\sqrt {\pi }}\right ) \sin \left (x^{2}+1\right )+\FresnelC \left (\frac {\sqrt {2}\, \left (x +1\right )}{\sqrt {\pi }}\right ) \sin \left (x^{2}+1\right )+\cos \left (x^{2}+1\right ) \left (\mathrm {S}\left (\frac {\sqrt {2}\, \left (x -1\right )}{\sqrt {\pi }}\right )-\mathrm {S}\left (\frac {\sqrt {2}\, \left (x +1\right )}{\sqrt {\pi }}\right )\right )\right ) \sqrt {2}\, \sqrt {\pi }}{2}-\cos \left (2 x \right )+1 \]
✓ Solution by Mathematica
Time used: 0.813 (sec). Leaf size: 113
DSolve[x*y''[x]-y'[x]+4*x^3*y[x]==8*x^3*Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {\frac {\pi }{2}} \left (\left (\text {FresnelC}\left (\sqrt {\frac {2}{\pi }} (x+1)\right )-\text {FresnelC}\left (\sqrt {\frac {2}{\pi }} (x-1)\right )\right ) \sin \left (x^2+1\right )+\left (S\left (\sqrt {\frac {2}{\pi }} (x-1)\right )-S\left (\sqrt {\frac {2}{\pi }} (x+1)\right )\right ) \cos \left (x^2+1\right )\right )+c_1 \cos \left (x^2\right )+c_2 \sin \left (x^2\right )+2 \sin ^2(x) \\ \end{align*}