58.2.23 problem 24

Internal problem ID [9146]
Book : Second order enumerated odes
Section : section 2
Problem number : 24
Date solved : Monday, January 27, 2025 at 05:49:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{3} y&=8 x^{3} \sin \left (x \right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.006 (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)
 

\[ y = \sin \left (x^{2}\right ) c_{2} +\cos \left (x^{2}\right ) c_{1} +1-\cos \left (2 x \right )-\frac {\sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (x -1\right )}{\sqrt {\pi }}\right ) \sqrt {2}\, \sin \left (x^{2}+1\right )}{2}+\frac {\sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (x -1\right )}{\sqrt {\pi }}\right ) \sqrt {2}\, \cos \left (x^{2}+1\right )}{2}+\frac {\sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (x +1\right )}{\sqrt {\pi }}\right ) \sqrt {2}\, \sin \left (x^{2}+1\right )}{2}-\frac {\sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (x +1\right )}{\sqrt {\pi }}\right ) \sqrt {2}\, \cos \left (x^{2}+1\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.417 (sec). Leaf size: 72

DSolve[x*D[y[x],{x,2}]-D[y[x],x]+4*x^3*y[x]==8*x^3*Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos \left (x^2\right ) \int _1^x-4 K[1] \sin ^2(K[1]) \sin \left (K[1]^2\right )dK[1]+\sin \left (x^2\right ) \int _1^x4 \cos \left (K[2]^2\right ) K[2] \sin ^2(K[2])dK[2]+c_1 \cos \left (x^2\right )+c_2 \sin \left (x^2\right ) \]