23.1.370 problem 355

Internal problem ID [4977]
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 : 355
Date solved : Sunday, October 12, 2025 at 01:20:23 AM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 15

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

\[ y = \arctan \left (\frac {\ln \left (x \right )-c_1}{x^{2}}\right ) \]

✓ Mathematica. Time used: 2.616 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to \arctan \left (\frac {\log (x)+4 c_1}{x^2}\right )\\ y(x)&\to -\frac {1}{2} \pi \sqrt {\frac {1}{x^4}} x^2\\ y(x)&\to \frac {1}{2} \pi \sqrt {\frac {1}{x^4}} x^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), x) - (-2*x**2*sin(y(x)) + cos(y(x)))*cos(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*sin(2*y(x)) + cos(2*y(x))/2 + 1/2)/x**3 cannot be solved by the factorable group method