65.10.3 problem 17.3

Internal problem ID [13715]
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 : 17.3
Date solved : Monday, March 31, 2025 at 08:09:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (t \cos \left (t \right )-\sin \left (t \right )\right ) x^{\prime \prime }-x^{\prime } t \sin \left (t \right )-x \sin \left (t \right )&=0 \end{align*}

Using reduction of order method given that one solution is

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

✗ Maple

ode:=(t*cos(t)-sin(t))*diff(diff(x(t),t),t)-diff(x(t),t)*t*sin(t)-x(t)*sin(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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