62.3.5 problem 6.5

Internal problem ID [15439]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 6. Laplace transforms. Problems page 172
Problem number : 6.5
Date solved : Thursday, October 02, 2025 at 10:14:11 AM
CAS classification : [_erf]

\begin{align*} x^{\prime \prime }+2 x^{\prime } t -4 x&=1 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.139 (sec). Leaf size: 9

ode:=diff(diff(x(t),t),t)+2*t*diff(x(t),t)-4*x(t) = 1; 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = \frac {t^{2}}{2} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 12

ode=D[x[t],{t,2}]+2*t*D[x[t],t]-4*x[t]==1; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {t^2}{2} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(2*t*Derivative(x(t), t) - 4*x(t) + Derivative(x(t), (t, 2)) - 1,0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

NotImplementedError : The given ODE Derivative(x(t), t) - (4*x(t) - Derivative(x(t), (t, 2)) + 1)/(2*t) cannot be solved by the factorable group method