80.1.4 problem 5

Internal problem ID [18463]
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, March 31, 2025 at 05:29:59 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: 0.049 (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);
 

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

✓ Mathematica. Time used: 0.112 (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]
 

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

✗ 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