75.20.5 problem 640

Internal problem ID [17060]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 640
Date solved : Thursday, March 13, 2025 at 09:12:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (\tan \left (x \right )-2 \cot \left (x \right )\right ) y^{\prime }+2 \cot \left (x \right )^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.292 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)+(tan(x)-2*cot(x))*diff(y(x),x)+2*cot(x)^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (x \right ) \left (\sin \left (x \right ) c_{2} +c_{1} \right ) \]

✓ Mathematica. Time used: 2.182 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+(Tan[x]-2*Cot[x])*D[y[x],x]+2*Cot[x]^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \sqrt {-\sin ^2(x)}-c_2 \sin ^2(x) \]

✗ Sympy

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

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