54.3.412 problem 1429

Internal problem ID [12707]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1429
Date solved : Wednesday, October 01, 2025 at 02:20:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

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

\[ y = \left (c_1 +c_2 +\left (c_1 -c_2 \right ) \cos \left (x \right )\right ) \csc \left (x \right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 25

ode=D[y[x],{x,2}] == Csc[x]^2*y[x] - Cot[x]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1-i c_2 \cos (x)}{\sqrt {\sin ^2(x)}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(y(x) - sin(x)**2*Derivative(y(x), (x, 2)))/(sin(x)*cos(x)) + Derivative(y(x), x) cannot be solved by the factorable group method