54.2.282 problem 860

Internal problem ID [12156]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 860
Date solved : Wednesday, October 01, 2025 at 01:01:54 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {-\sin \left (2 y\right )+x \cos \left (2 y\right )+\cos \left (2 y\right ) x^{3}+\cos \left (2 y\right ) x^{4}+x +x^{3}+x^{4}}{2 x} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 27

ode:=diff(y(x),x) = 1/2*(-sin(2*y(x))+x*cos(2*y(x))+cos(2*y(x))*x^3+cos(2*y(x))*x^4+x+x^3+x^4)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arctan \left (\frac {4 x^{5}+5 x^{4}+10 x^{2}+c_1}{20 x}\right ) \]

✓ Mathematica. Time used: 1.64 (sec). Leaf size: 69

ode=D[y[x],x] == (x/2 + x^3/2 + x^4/2 + (x*Cos[2*y[x]])/2 + (x^3*Cos[2*y[x]])/2 + (x^4*Cos[2*y[x]])/2 - Sin[2*y[x]]/2)/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \arctan \left (\frac {x^4}{5}+\frac {x^3}{4}+\frac {x}{2}+\frac {c_1}{2 x}\right )\\ y(x)&\to -\frac {1}{2} \pi \sqrt {\frac {1}{x^2}} x\\ y(x)&\to \frac {1}{2} \pi \sqrt {\frac {1}{x^2}} x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**4*cos(2*y(x)) + x**4 + x**3*cos(2*y(x)) + x**3 + x*cos(2*y(x)) + x - sin(2*y(x)))/(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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