63.13.4 problem 5

Internal problem ID [13096]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises page 125
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 09:17:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} x&=t \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 12

ode:=diff(diff(x(t),t),t)-(2+t)/t*diff(x(t),t)+(2+t)/t^2*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = t \left (c_{1} +c_{2} {\mathrm e}^{t}\right ) \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 17

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

\[ x(t)\to e t \left (c_2 e^t+c_1\right ) \]

✓ Sympy. Time used: 0.827 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = C_{2} t^{2} \left (\frac {t^{3}}{24} + \frac {t^{2}}{6} + \frac {t}{2} + 1\right ) + C_{1} t + O\left (t^{6}\right ) \]