7.20.4 problem 32

Internal problem ID [558]
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 : 32
Date solved : Saturday, March 29, 2025 at 04:56:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.099 (sec). Leaf size: 20

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

\[ x = \frac {c_1 \,{\mathrm e}^{-t} \left (t \cosh \left (t \right )-\sinh \left (t \right )\right )}{2} \]

✓ Mathematica. Time used: 0.142 (sec). Leaf size: 46

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

\[ x(t)\to -\frac {\sqrt {\frac {2}{\pi }} c_1 e^{-t} \sqrt {t} (t \cosh (t)-\sinh (t))}{\sqrt {-i t}} \]

✓ Sympy. Time used: 0.758 (sec). Leaf size: 20

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

\[ x{\left (t \right )} = C_{1} t^{3} \left (\frac {3 t^{2}}{5} - t + 1\right ) + O\left (t^{6}\right ) \]