1.20.5 problem 33

Internal problem ID [559]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.4 (Derivatives, Integrals and products of transforms). Problems at page 303
Problem number : 33
Date solved : Tuesday, September 30, 2025 at 10:55:32 AM
CAS classification : [_Lienard]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 15

ode:=t*diff(diff(x(t),t),t)-2*diff(x(t),t)+t*x(t) = 0; 
ic:=[x(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 27

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

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

✗ Sympy

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

ValueError : Couldnt solve for initial conditions