90.23.5 problem 9

Internal problem ID [25358]
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 : 9
Date solved : Friday, October 03, 2025 at 12:00:36 AM
CAS classification : [_Lienard]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 21

ode:=t*diff(diff(y(t),t),t)-4*diff(y(t),t)+t*y(t) = 0; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\left (t^{2}-3\right ) c_1 \sin \left (t \right )}{8}-\frac {3 c_1 \cos \left (t \right ) t}{8} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to -\sqrt {\frac {2}{\pi }} c_1 \left (\left (t^2-3\right ) \sin (t)+3 t \cos (t)\right ) \end{align*}

✗ Sympy

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

ValueError : Couldnt solve for initial conditions