67.2.15 problem Problem 1(o)
Internal
problem
ID
[13901]
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
:
Monday, March 31, 2025 at 08:17:30 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} y^{\prime \prime }-y&=\sin \left (x \right )^{2} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 97
ode:=x^2*diff(diff(y(x),x),x)-y(x) = sin(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {3 \left (\sqrt {5}+\frac {5}{3}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, -\frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}-\frac {\sqrt {5}}{4}\right ], -x^{2}\right )}{10}-\frac {3 \left (\sqrt {5}-\frac {5}{3}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, \frac {\sqrt {5}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}, 2, \frac {7}{4}+\frac {\sqrt {5}}{4}\right ], -x^{2}\right )}{10}+x^{-\frac {\sqrt {5}}{2}+\frac {1}{2}} c_1 +x^{\frac {\sqrt {5}}{2}+\frac {1}{2}} c_2
\]
✓ Mathematica. Time used: 0.501 (sec). Leaf size: 349
ode=x^2*D[y[x],{x,2}]-y[x]==Sin[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {x^{-\frac {\sqrt {5}}{2}} \left (20 \left (1+\sqrt {5}\right ) \sqrt {x} \int _1^x-\frac {K[1]^{\frac {1}{2} \left (-3+\sqrt {5}\right )} \sin ^2(K[1])}{\sqrt {5}}dK[1]+20 \sqrt {5} c_2 x^{\frac {1}{2}+\sqrt {5}}+20 c_2 x^{\frac {1}{2}+\sqrt {5}}-4 \sqrt {5} x^{\frac {\sqrt {5}}{2}}+2^{\frac {1}{2} \left (1+\sqrt {5}\right )} \left (5+\sqrt {5}\right ) x^{\frac {\sqrt {5}}{2}} (-i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},-2 i x\right )+2^{\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {5} (i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} x^{\frac {\sqrt {5}}{2}} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},2 i x\right )+5\ 2^{\frac {1}{2} \left (1+\sqrt {5}\right )} (i x)^{\frac {1}{2} \left (1+\sqrt {5}\right )} x^{\frac {\sqrt {5}}{2}} \Gamma \left (-\frac {1}{2}-\frac {\sqrt {5}}{2},2 i x\right )+20 \sqrt {5} c_1 \sqrt {x}+20 c_1 \sqrt {x}\right )}{20 \left (1+\sqrt {5}\right )}
\]
✓ Sympy. Time used: 12.503 (sec). Leaf size: 957
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) - y(x) - sin(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\text {Solution too large to show}
\]