60.2.390 problem 968

Internal problem ID [10964]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 968
Date solved : Sunday, March 30, 2025 at 07:33:46 PM
CAS classification : [[_homogeneous, `class D`]]

\begin{align*} y^{\prime }&=\frac {-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 23

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

\[ y = \frac {\arccos \left (1+c_1 \left (x +1\right )^{2} {\mathrm e}^{x \left (x -2\right )}\right ) x}{2} \]

✓ Mathematica. Time used: 46.919 (sec). Leaf size: 36

ode=D[y[x],x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sec[y[x]/x]*(x^4*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*Sin[y[x]/x] + (Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*y[x])/2 + (x*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*y[x])/2 - (Sin[y[x]/x]*y[x])/2 - (x*Sin[y[x]/x]*y[x])/2 + (Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2 + (x*Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2))/(x*(1 + x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to x \arcsin \left (\exp \left (\int _1^x\frac {K[1]^2}{K[1]+1}dK[1]+c_1\right )\right ) \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x**4*sin(y(x)/(2*x))*sin(y(x)/x)*cos(y(x)/(2*x)) + x*y(x)*sin(y(x)/(2*x))*cos(y(x)/(2*x)) - x*y(x)*sin(y(x)/x) + x*y(x)*sin(3*y(x)/(2*x))*cos(y(x)/(2*x)) + y(x)*sin(y(x)/(2*x))*cos(y(x)/(2*x)) - y(x)*sin(y(x)/x) + y(x)*sin(3*y(x)/(2*x))*cos(y(x)/(2*x)))/(2*x*(x + 1)*sin(y(x)/(2*x))*cos(y(x)/(2*x))*cos(y(x)/x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

PolynomialDivisionFailed : couldnt reduce degree in a polynomial division algorithm when dividing [[], []] by [[ANP([mpq(1,1)], [mpq(1,1), mpq(0,1), mpq(1,1)], QQ), ANP([mpq(-2,1), mpq(0,1)], [mpq(1,1), mpq(0,1), mpq(1,1)], QQ), ANP([mpq(-2,1)], [mpq(1,1), mpq(0,1), mpq(1,1)], QQ)]]. This can happen when its not possible to detect zero in the coefficient domain. The domain of computation is QQ<I>. Zero detection is guaranteed in this coefficient domain. This may indicate a bug in SymPy or the domain is user defined and doesnt implement zero detection properly.