67.2.15 problem Problem 1(o)
Internal
problem
ID
[13972]
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)
Date
solved
:
Tuesday, January 28, 2025 at 06:10:40 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} y^{\prime \prime }-y&=\sin \left (x \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 97
dsolve(x^2*diff(y(x),x$2)-y(x)=sin(x)^2,y(x), singsol=all)
\[
y = \frac {3 x^{2} \left (\sqrt {5}+\frac {5}{3}\right ) \operatorname {hypergeom}\left (\left [1, -\frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}-\frac {\sqrt {5}}{4}\right ], -x^{2}\right )}{10}-\frac {3 x^{2} \left (\sqrt {5}-\frac {5}{3}\right ) \operatorname {hypergeom}\left (\left [1, \frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}+\frac {\sqrt {5}}{4}\right ], -x^{2}\right )}{10}+x^{\frac {1}{2}-\frac {\sqrt {5}}{2}} c_{1} +x^{\frac {1}{2}+\frac {\sqrt {5}}{2}} c_{2}
\]
✓ Solution by Mathematica
Time used: 0.501 (sec). Leaf size: 349
DSolve[x^2*D[y[x],{x,2}]-y[x]==Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {x^{-\frac {\sqrt {5}}{2}} \left (20 \left (1+\sqrt {5}\right ) \sqrt {x} \int _1^x-\frac {K[1]^{\frac {1}{2} \left (-3+\sqrt {5}\right )} \sin ^2(K[1])}{\sqrt {5}}dK[1]+20 \sqrt {5} c_2 x^{\frac {1}{2}+\sqrt {5}}+20 c_2 x^{\frac {1}{2}+\sqrt {5}}-4 \sqrt {5} x^{\frac {\sqrt {5}}{2}}+2^{\frac {1}{2} \left (1+\sqrt {5}\right )} \left (5+\sqrt {5}\right ) x^{\frac {\sqrt {5}}{2}} (-i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},-2 i x\right )+2^{\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {5} (i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} x^{\frac {\sqrt {5}}{2}} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},2 i x\right )+5\ 2^{\frac {1}{2} \left (1+\sqrt {5}\right )} (i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} x^{\frac {\sqrt {5}}{2}} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},2 i x\right )+20 \sqrt {5} c_1 \sqrt {x}+20 c_1 \sqrt {x}\right )}{20 \left (1+\sqrt {5}\right )}
\]