77.38.3 problem 3

Internal problem ID [20746]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 3
Date solved : Thursday, October 02, 2025 at 06:22:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

ode:=(x^3-x)*diff(diff(y(x),x),x)+diff(y(x),x)+n^2*x^3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sin \left (n \sqrt {x^{2}-1}\right )+c_2 \cos \left (n \sqrt {x^{2}-1}\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 36

ode=(x^3-x)*D[y[x],{x,2}]+D[y[x],x]+n^2*x^3*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos \left (n \sqrt {x^2-1}\right )+c_2 \sin \left (n \sqrt {x^2-1}\right ) \end{align*}

✗ Sympy

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

False