29.12.33 problem 352

Internal problem ID [4952]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 12
Problem number : 352
Date solved : Friday, March 14, 2025 at 01:28:51 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: 0.012 (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);
 

\[ y \left (x \right ) = \arctan \left (\frac {\ln \left (x \right )-c_{1}}{x^{2}}\right ) \]

✓ Mathematica. Time used: 4.357 (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