Internal problem ID [11916]
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]]
\[ \boxed {x^{2} y^{\prime \prime }-y=\sin \left (x \right )^{2}} \]
✓ 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: 1.679 (sec). Leaf size: 445
DSolve[x^2*y''[x]-y[x]==Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {10 \sqrt {5} c_1 x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}+10 c_1 x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}+10 \sqrt {5} c_2 x^{\frac {1}{2} \left (1+\sqrt {5}\right )}+10 c_2 x^{\frac {1}{2} \left (1+\sqrt {5}\right )}+2^{\frac {1}{2} \left (\sqrt {5}-1\right )} \left (5+\sqrt {5}\right ) (-i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},-2 i x\right )-2^{-\frac {1}{2}-\frac {\sqrt {5}}{2}} \sqrt {5} (-i x)^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \Gamma \left (\frac {1}{2} \left (-1+\sqrt {5}\right ),-2 i x\right )-5\ 2^{-\frac {1}{2}-\frac {\sqrt {5}}{2}} (-i x)^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \Gamma \left (\frac {1}{2} \left (-1+\sqrt {5}\right ),-2 i x\right )+2^{\frac {1}{2} \left (\sqrt {5}-1\right )} \left (5+\sqrt {5}\right ) (i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},2 i x\right )-2^{-\frac {1}{2}-\frac {\sqrt {5}}{2}} \sqrt {5} (i x)^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \Gamma \left (\frac {1}{2} \left (-1+\sqrt {5}\right ),2 i x\right )-5\ 2^{-\frac {1}{2}-\frac {\sqrt {5}}{2}} (i x)^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \Gamma \left (\frac {1}{2} \left (-1+\sqrt {5}\right ),2 i x\right )-5 \sqrt {5}-5}{10 \left (1+\sqrt {5}\right )} \]