65.10.6 problem 17.6

Internal problem ID [13710]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 17, Reduction of order. Exercises page 162
Problem number : 17.6
Date solved : Wednesday, March 05, 2025 at 10:13:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.367 (sec). Leaf size: 13

ode:=tan(t)*diff(diff(x(t),t),t)-3*diff(x(t),t)+(tan(t)+3*cot(t))*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = \sin \left (t \right ) \left (c_{1} +c_{2} \cos \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.187 (sec). Leaf size: 24

ode=Tan[t]*D[x[t],{t,2}]-3*D[x[t],t]+(Tan[t]+3*Cot[t])*x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \sqrt {-\sin ^2(t)} (c_2 \cos (t)+c_1) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((tan(t) + 3/tan(t))*x(t) + tan(t)*Derivative(x(t), (t, 2)) - 3*Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

NotImplementedError : The given ODE -((x(t) + Derivative(x(t), (t, 2)))*tan(t)**2/3 + x(t))/tan(t) + Derivative(x(t), t) cannot be solved by the factorable group method