2.1.4 Problem 5

Internal problem ID [19705]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter 1. section 5. Problems at page 19
Problem number : 5
Date solved : Monday, December 01, 2025 at 03:23:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 37

ode:=cos(x)*diff(y(x),x)+sin(x)*diff(diff(y(x),x),x)+n*y(x)*sin(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {LegendreP}\left (\frac {\sqrt {4 n +1}}{2}-\frac {1}{2}, \cos \left (x \right )\right )+c_2 \operatorname {LegendreQ}\left (\frac {\sqrt {4 n +1}}{2}-\frac {1}{2}, \cos \left (x \right )\right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x),\ 
 dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
   trying a quadrature 
   checking if the LODE has constant coefficients 
   checking if the LODE is of Euler type 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying a Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      <- Legendre successful 
   <- special function solution successful 
   Change of variables used: 
      [x = arccos(t)] 
   Linear ODE actually solved: 
      (-n*t^2+n)*u(t)+(2*t^3-2*t)*diff(u(t),t)+(t^4-2*t^2+1)*diff(diff(u(t),t),\ 
t) = 0 
<- change of variables successful
 

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 48

ode=D[Sin[x]*D[y[x],x],x]+n*y[x]*Sin[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \operatorname {LegendreP}\left (\frac {1}{2} \left (\sqrt {4 n+1}-1\right ),\cos (x)\right )+c_2 \operatorname {LegendreQ}\left (\frac {1}{2} \left (\sqrt {4 n+1}-1\right ),\cos (x)\right ) \end{align*}

✗ Sympy

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

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