54.2.97 problem 673

Internal problem ID [11971]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 673
Date solved : Tuesday, September 30, 2025 at 11:50:49 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 17

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

\[ y = \arctan \left (\frac {x^{3}+6 c_1}{3 x}\right ) \]

✓ Mathematica. Time used: 0.948 (sec). Leaf size: 57

ode=D[y[x],x] == (x^2/2 + (x^2*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 {2 x^3+3 c_1}{6 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**2*cos(2*y(x)) + x**2 - sin(2*y(x)))/(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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