90.22.14 problem 14

Internal problem ID [25352]
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 : 14
Date solved : Friday, October 03, 2025 at 12:00:31 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} y^{\prime \prime }+t y^{\prime }+4 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-3 \\ y^{\prime }\left (1\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 19

ode:=t^2*diff(diff(y(t),t),t)+t*diff(y(t),t)+4*y(t) = 0; 
ic:=[y(1) = -3, D(y)(1) = 4]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 2 \sin \left (2 \ln \left (t \right )\right )-3 \cos \left (2 \ln \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 20

ode=t^2*D[y[t],{t,2}]+t*D[y[t],t]+4*y[t]==0; 
ic={y[1]==-3,Derivative[1][y][1] ==4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 \sin (2 \log (t))-3 \cos (2 \log (t)) \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + t*Derivative(y(t), t) + 4*y(t),0) 
ics = {y(1): -3, Subs(Derivative(y(t), t), t, 1): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 \sin {\left (2 \log {\left (t \right )} \right )} - 3 \cos {\left (2 \log {\left (t \right )} \right )} \]