7.20.6 problem 34

Internal problem ID [560]
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 : 34
Date solved : Tuesday, March 04, 2025 at 11:26:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t x^{\prime \prime }+\left (4 t -2\right ) x^{\prime }+\left (13 t -4\right ) x&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.206 (sec). Leaf size: 24

ode:=t*diff(diff(x(t),t),t)+(4*t-2)*diff(x(t),t)+(13*t-4)*x(t) = 0; 
ic:=x(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = -\frac {{\mathrm e}^{-2 t} c_1 \left (\cos \left (3 t \right ) t -\frac {\sin \left (3 t \right )}{3}\right )}{18} \]

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 41

ode=t*D[x[t],{t,2}]+(4*t-2)*D[x[t],t]+(13*t-4)*x[t]==0; 
ic={x[0]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\frac {1}{3} \sqrt {\frac {2}{3 \pi }} c_1 e^{-2 t} (3 t \cos (3 t)-\sin (3 t)) \]

✓ Sympy. Time used: 0.851 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t*Derivative(x(t), (t, 2)) + (4*t - 2)*Derivative(x(t), t) + (13*t - 4)*x(t),0) 
ics = {x(0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} t^{3} \left (\frac {11 t^{2}}{10} - 2 t + 1\right ) + O\left (t^{6}\right ) \]