54.3.400 problem 1417

Internal problem ID [12695]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1417
Date solved : Wednesday, October 01, 2025 at 02:20:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 31

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

\[ y = {\mathrm e}^{\frac {\cos \left (x \right )}{2}} \left (c_1 \sin \left (\frac {\sqrt {3}\, \cos \left (x \right )}{2}\right )+c_2 \cos \left (\frac {\sqrt {3}\, \cos \left (x \right )}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.101 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to e^{\frac {\cos (x)}{2}} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \cos (x)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \cos (x)\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((sin(x)**2 - cos(x))*Derivative(y(x), x)/sin(x) + y(x)*sin(x)**2 + Derivative(y(x), (x, 2)),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)/(-sin(x)**2 + cos(x)) + Derivative(y(x), x) cannot be solved by the factorable group method