2.15 problem Problem 1(o)

Internal problem ID [10888]

Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number: Problem 1(o).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

\[ \boxed {x^{2} y^{\prime \prime }-y-\sin \left (x \right )^{2}=0} \]

Solution by Maple

Time used: 0.016 (sec). Leaf size: 147

dsolve(x^2*diff(y(x),x$2)-y(x)=sin(x)^2,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{2} x^{\frac {\sqrt {5}}{2}+\frac {1}{2}}+c_{1} x^{-\frac {\sqrt {5}}{2}+\frac {1}{2}}+\frac {x^{2} \left (3 \operatorname {hypergeom}\left (\left [1, -\frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}-\frac {\sqrt {5}}{4}\right ], -x^{2}\right ) \sqrt {5}-3 \operatorname {hypergeom}\left (\left [1, \frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}+\frac {\sqrt {5}}{4}\right ], -x^{2}\right ) \sqrt {5}+5 \operatorname {hypergeom}\left (\left [1, -\frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}-\frac {\sqrt {5}}{4}\right ], -x^{2}\right )+5 \operatorname {hypergeom}\left (\left [1, \frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}+\frac {\sqrt {5}}{4}\right ], -x^{2}\right )\right )}{10} \]

Solution by Mathematica

Time used: 0.539 (sec). Leaf size: 129

DSolve[x^2*y''[x]-y[x]==Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-\operatorname {ExpIntegralE}\left (\frac {3}{2}-\frac {\sqrt {5}}{2},-2 i x\right )-\operatorname {ExpIntegralE}\left (\frac {3}{2}-\frac {\sqrt {5}}{2},2 i x\right )+\operatorname {ExpIntegralE}\left (\frac {1}{2} \left (3+\sqrt {5}\right ),-2 i x\right )+\operatorname {ExpIntegralE}\left (\frac {1}{2} \left (3+\sqrt {5}\right ),2 i x\right )}{4 \sqrt {5}}+c_2 x^{\frac {1}{2} \left (1+\sqrt {5}\right )}+c_1 x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}-\frac {1}{2} \\ \end{align*}