60.3.410 problem 1427

Internal problem ID [11406]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1427
Date solved : Sunday, March 30, 2025 at 08:20:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}} \end{align*}

✓ Maple. Time used: 0.358 (sec). Leaf size: 88

ode:=diff(diff(y(x),x),x) = -(-(a^2*b^2-(a+1)^2)*sin(x)^2-a*(a+1)*b*sin(2*x)-a*(a-1))/sin(x)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\cot \left (x \right )-i\right )^{-\frac {1}{2}+\frac {1}{2} i a b -\frac {1}{2} a} \left (c_1 \left (\cot \left (x \right )+i\right )^{-\frac {1}{2}-\frac {1}{2} a -\frac {1}{2} i a b} \left (b +\cot \left (x \right )\right )+c_2 \left (\cot \left (x \right )+i\right )^{\frac {1}{2}+\frac {1}{2} a +\frac {1}{2} i a b} \operatorname {hypergeom}\left (\left [a \left (i b +1\right ), i a b -a +1\right ], \left [i a b +a +2\right ], \frac {1}{2}-\frac {i \cot \left (x \right )}{2}\right )\right ) \]

✓ Mathematica. Time used: 1.741 (sec). Leaf size: 94

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

\[ y(x)\to c_2 \left ((2 a+1) e^{a b x} \sin ^a(x) (b \sin (x)+\cos (x)) \int _1^xe^{-2 a b K[1]} \sin ^{-2 (a+1)}(K[1])dK[1]+e^{-a b x} \sin ^{-a-1}(x)\right )+c_1 e^{a b x} \sin ^a(x) (b \sin (x)+\cos (x)) \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve (-a*b*(a + 1)*sin(2*x) - a*(a - 1) + (-a**2*b**2 + (a + 1)**2)*sin(x)**2)*y(x)/sin(x)**2 + Derivative(y(x), (x, 2))