60.3.408 problem 1425

Internal problem ID [11404]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1425
Date solved : Sunday, March 30, 2025 at 08:20:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \end{align*}

✓ Maple. Time used: 0.409 (sec). Leaf size: 83

ode:=diff(diff(y(x),x),x) = -(-a^2*cos(x)^2-(3-2*a)*cos(x)-3+3*a)/sin(x)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.003 (sec). Leaf size: 125

ode=D[y[x],{x,2}] == (3 - 3*a + (3 - 2*a)*Cos[x] + a^2*Cos[x]^2)*Csc[x]^2*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int -\frac {\csc ^2(x) \left (a (2 a-1) \cos ^2(x)+(3-4 a) \cos (x)-2 a+3\right )}{(2 a-1) \cos (x)-2} \, d\cos (x)\right ) \left (c_2 \int _1^{\cos (x)}\exp \left (\int _1^{K[2]}\frac {-4 a^2 K[1]^2+K[1]^2-4 K[1]+a (8 K[1]+4)-6}{((2 a-1) K[1]-2) \left (K[1]^2-1\right )}dK[1]\right )dK[2]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq((-a**2*cos(x)**2 + 3*a - (3 - 2*a)*cos(x) - 3)*y(x)/sin(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve (-a**2*cos(x)**2 + 3*a - (3 - 2*a)*cos(x) - 3)*y(x)/sin(x)**2 + Derivative(y(x), (x, 2))