54.3.420 problem 1437

Internal problem ID [12715]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1437
Date solved : Wednesday, October 01, 2025 at 02:21:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}+\frac {\sin \left (x \right )^{2} y}{\cos \left (x \right )^{2}} \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x) = (3*sin(x)^2+1)/cos(x)/sin(x)*diff(y(x),x)+sin(x)^2/cos(x)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \cos \left (x \right )^{-\frac {3}{2}+\frac {\sqrt {13}}{2}}+c_2 \cos \left (x \right )^{-\frac {3}{2}-\frac {\sqrt {13}}{2}} \]

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 36

ode=D[y[x],{x,2}] == Tan[x]^2*y[x] + Csc[x]*Sec[x]*(1 + 3*Sin[x]^2)*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos ^{-\frac {3}{2}-\frac {\sqrt {13}}{2}}(x) \left (c_2 \cos ^{\sqrt {13}}(x)+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-3*sin(x)**2 - 1)*Derivative(y(x), x)/(sin(x)*cos(x)) - y(x)*sin(x)**2/cos(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 + cos(x)**2*Derivative(y(x), (x, 2)))*tan(x)/(3*sin(x)**2 + 1) + Derivative(y(x), x) cannot be solved by the factorable group method