54.3.414 problem 1431

Internal problem ID [12709]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1431
Date solved : Friday, October 03, 2025 at 03:46:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.151 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.572 (sec). Leaf size: 119

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

\begin{align*} y(x)&\to \frac {1}{2} \cos (2 x) \exp \left (\int _1^{\cos (x)}-\frac {1}{4 K[1]-4 K[1]^3}dK[1]-\frac {1}{2} \int _1^{\cos (x)}-\frac {1}{2 K[2]-2 K[2]^3}dK[2]\right ) \left (c_2 \int _1^{\cos (x)}\frac {4 \exp \left (-2 \int _1^{K[3]}-\frac {1}{4 K[1]-4 K[1]^3}dK[1]\right )}{\left (1-2 K[3]^2\right )^2}dK[3]+c_1\right ) \end{align*}

✗ Sympy

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

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