7.20.3 problem 31

Internal problem ID [557]
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 : 31
Date solved : Saturday, March 29, 2025 at 04:56:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t x^{\prime \prime }-\left (4 t +1\right ) x^{\prime }+2 \left (2 t +1\right ) x&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.089 (sec). Leaf size: 14

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

\[ x = \frac {c_1 \,t^{2} {\mathrm e}^{2 t}}{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 19

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

\[ x(t)\to \frac {1}{2} c_2 e^{2 t} t^2 \]

✓ Sympy. Time used: 0.845 (sec). Leaf size: 27

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

\[ x{\left (t \right )} = C_{1} t^{2} \left (\frac {4 t^{3}}{3} + 2 t^{2} + 2 t + 1\right ) + O\left (t^{6}\right ) \]