90.24.15 problem 15

Internal problem ID [25380]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 371
Problem number : 15
Date solved : Friday, October 03, 2025 at 12:00:45 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-t^{2}+1\right ) y^{\prime \prime }+2 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=-t^{2}+1 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 43

ode:=(-t^2+1)*diff(diff(y(t),t),t)+2*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (-t^{2}+1\right ) c_1 \ln \left (t -1\right )}{4}+\frac {c_1 \left (t^{2}-1\right ) \ln \left (t +1\right )}{4}+c_2 \,t^{2}-\frac {c_1 t}{2}-c_2 \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 90

ode=(1-t^2)*D[y[t],{t,2}]-2*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to i c_2 t \operatorname {Hypergeometric2F1}\left (\frac {1}{4}-\frac {i \sqrt {7}}{4},\frac {1}{4}+\frac {i \sqrt {7}}{4},\frac {3}{2},t^2\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4} i \left (-i+\sqrt {7}\right ),\frac {1}{4} i \left (i+\sqrt {7}\right ),\frac {1}{2},t^2\right ) \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 80

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((1 - t**2)*Derivative(y(t), (t, 2)) - 2*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\left (C_{1} \sqrt {t^{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4} - \frac {\sqrt {7} i}{4}, \frac {1}{4} + \frac {\sqrt {7} i}{4} \\ \frac {3}{2} \end {matrix}\middle | {t^{2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4} - \frac {\sqrt {7} i}{4}, - \frac {1}{4} + \frac {\sqrt {7} i}{4} \\ \frac {1}{2} \end {matrix}\middle | {t^{2}} \right )}\right ) \sqrt [4]{t^{2}}}{\sqrt {t}} \]