23.1.209 problem 205

Internal problem ID [4816]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 205
Date solved : Tuesday, September 30, 2025 at 08:42:32 AM
CAS classification : [_separable]

\begin{align*} x y^{\prime }&=\left (-2 x^{2}+1\right ) \cot \left (y\right )^{2} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 22

ode:=x*diff(y(x),x) = (-2*x^2+1)*cot(y(x))^2; 
dsolve(ode,y(x), singsol=all);
 

\[ x^{2}-\ln \left (x \right )+\tan \left (y\right )+\frac {\pi }{2}-y+c_1 = 0 \]

✓ Mathematica. Time used: 0.332 (sec). Leaf size: 55

ode=x*D[y[x],x]==(1-2*x^2)*Cot[y[x]]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{2} (\tan (\text {$\#$1})-\arctan (\tan (\text {$\#$1})))\&\right ]\left [-\frac {x^2}{2}+\frac {\log (x)}{2}+c_1\right ]\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 3.756 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - (1 - 2*x**2)/tan(y(x))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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