Internal problem ID [6707]
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]]
\[ \boxed {x y^{\prime \prime }-y^{\prime }+4 y x^{3}-8 x^{3} \sin \left (x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 124
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)
\[ y \left (x \right ) = \sin \left (x^{2}\right ) c_{2} +\cos \left (x^{2}\right ) c_{1} +1-\cos \left (2 x \right )-\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (x -1\right )}{\sqrt {\pi }}\right ) \sin \left (x^{2}+1\right )}{2}+\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (x -1\right )}{\sqrt {\pi }}\right ) \cos \left (x^{2}+1\right )}{2}+\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (x +1\right )}{\sqrt {\pi }}\right ) \sin \left (x^{2}+1\right )}{2}-\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (x +1\right )}{\sqrt {\pi }}\right ) \cos \left (x^{2}+1\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.492 (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 (\operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} (x+1)\right )-\operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} (x-1)\right )\right ) \sin \left (x^2+1\right )+\left (\operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} (x-1)\right )-\operatorname {FresnelS}\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*}