90.22.12 problem 12

Internal problem ID [25350]
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 353
Problem number : 12
Date solved : Friday, October 03, 2025 at 12:00:30 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }+2 t y^{\prime }-2 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 13

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

\[ y = -\frac {1}{3 t^{2}}+\frac {t}{3} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 17

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

\begin{align*} y(t)&\to \frac {t^3-1}{3 t^2} \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 12

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

\[ y{\left (t \right )} = \frac {t}{3} - \frac {1}{3 t^{2}} \]