67.18.14 problem 27.4

Internal problem ID [16885]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.4
Date solved : Thursday, October 02, 2025 at 01:40:07 PM
CAS classification : [_Lienard]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.082 (sec). Leaf size: 7

ode:=t*diff(diff(y(t),t),t)+diff(y(t),t)+t*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \operatorname {BesselJ}\left (0, t\right ) \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 8

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

\begin{align*} y(t)&\to \operatorname {BesselJ}(0,t) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 5

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

\[ y{\left (t \right )} = J_{0}\left (t\right ) \]