54.3.403 problem 1420

Internal problem ID [12698]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1420
Date solved : Friday, October 03, 2025 at 03:46:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.233 (sec). Leaf size: 110

ode:=cos(x)^2*diff(diff(y(x),x),x)-(a*cos(x)^2+n*(n-1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sin \left (x \right )^{{3}/{2}} \left (\cos \left (x \right )^{\frac {1}{2}+n} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}+n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_1 +\left (\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}-\frac {n}{2}} \operatorname {hypergeom}\left (\left [1+\frac {i \sqrt {a}}{2}-\frac {n}{2}, 1-\frac {i \sqrt {a}}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}-n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_2 \right )}{\sqrt {\sin \left (2 x \right )}} \]

✓ Mathematica. Time used: 0.348 (sec). Leaf size: 126

ode=(-((-1 + n)*n) - a*Cos[x]^2)*y[x] + Cos[x]^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 i^{1-n} \cos ^{1-n}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-n-i \sqrt {a}+1\right ),\frac {1}{2} \left (-n+i \sqrt {a}+1\right ),\frac {3}{2}-n,\cos ^2(x)\right )+c_2 i^n \cos ^n(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (n-i \sqrt {a}\right ),\frac {1}{2} \left (n+i \sqrt {a}\right ),n+\frac {1}{2},\cos ^2(x)\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq((-a*cos(x)**2 - n*(n - 1))*y(x) + cos(x)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False