90.24.7 problem 7

Internal problem ID [25372]
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 : 7
Date solved : Friday, October 03, 2025 at 12:00:41 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t y^{\prime \prime }-y^{\prime }+4 t^{3} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\sin \left (t^{2}\right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=t*diff(diff(y(t),t),t)-diff(y(t),t)+4*t^3*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_1 \sin \left (t^{2}\right )+c_2 \cos \left (t^{2}\right ) \]

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

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

\begin{align*} y(t)&\to c_1 \cos \left (t^2\right )+c_2 \sin \left (t^2\right ) \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = t \left (C_{1} J_{\frac {1}{2}}\left (t^{2}\right ) + C_{2} Y_{\frac {1}{2}}\left (t^{2}\right )\right ) \]