62.36.9 problem Ex 10

Internal problem ID [12936]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 10
Date solved : Monday, March 31, 2025 at 07:27:28 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 21

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

\[ y = -\frac {{\mathrm e}^{\frac {c_1}{2}} \operatorname {Ei}_{1}\left (\ln \left (\tan \left (x \right )\right )+\frac {c_1}{2}\right )}{2}+c_2 \]

✗ Mathematica

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

Not solved

✓ Sympy. Time used: 83.322 (sec). Leaf size: 58

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

\[ \left [ y{\left (x \right )} = C_{1} + \int \frac {1}{\left (C_{2} + 2 \log {\left (\sin {\left (x \right )} \right )} - \log {\left (- \cos ^{2}{\left (x \right )} \right )}\right ) \sin ^{2}{\left (x \right )}}\, dx, \ y{\left (x \right )} = C_{1} + \int \frac {1}{\left (C_{2} + 2 \log {\left (\sin {\left (x \right )} \right )} - \log {\left (- \cos ^{2}{\left (x \right )} \right )}\right ) \sin ^{2}{\left (x \right )}}\, dx\right ] \]