54.3.411 problem 1428

Internal problem ID [12706]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1428
Date solved : Friday, October 03, 2025 at 03:46:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}} \end{align*}

✓ Maple. Time used: 0.258 (sec). Leaf size: 161

ode:=diff(diff(y(x),x),x) = -(a*cos(x)^2+b*sin(x)^2+c)/sin(x)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.302 (sec). Leaf size: 87

ode=D[y[x],{x,2}] == -(Csc[x]^2*(c + a*Cos[x]^2 + b*Sin[x]^2)*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{\sqrt {b-a}-\frac {1}{2}}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))+c_2 Q_{\sqrt {b-a}-\frac {1}{2}}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))\right ) \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve (a*cos(x)**2 + b*sin(x)**2 + c)*y(x)/sin(x)**2 + Derivative(y(x), (x, 2))