90.23.4 problem 8

Internal problem ID [25357]
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 365
Problem number : 8
Date solved : Friday, October 03, 2025 at 12:00:35 AM
CAS classification : [_Lienard]

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

Using Laplace method

✓ Maple. Time used: 0.044 (sec). Leaf size: 28

ode:=t*diff(diff(y(t),t),t)-2*diff(y(t),t)+t*y(t) = 0; 
dsolve(ode,y(t),method='laplace');
 

\[ y = -\frac {\cos \left (t \right ) \left (t c_1 -2 y \left (0\right )\right )}{2}+\frac {\sin \left (t \right ) \left (2 t y \left (0\right )+c_1 \right )}{2} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 39

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

\begin{align*} y(t)&\to -\sqrt {\frac {2}{\pi }} ((c_1 t+c_2) \cos (t)+(c_2 t-c_1) \sin (t)) \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = t^{\frac {3}{2}} \left (C_{1} J_{\frac {3}{2}}\left (t\right ) + C_{2} Y_{\frac {3}{2}}\left (t\right )\right ) \]