54.2.243 problem 820

Internal problem ID [12117]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 820
Date solved : Sunday, October 12, 2025 at 02:21:00 AM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 27

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

\[ y = \operatorname {arcsec}\left (\frac {2 x^{2} \ln \left (x \right )-x^{2}+4 c_1}{4 \ln \left (x \right )}\right ) \]

✓ Mathematica. Time used: 0.827 (sec). Leaf size: 77

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

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

✓ Sympy. Time used: 3.065 (sec). Leaf size: 49

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

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (- \frac {4 \log {\left (x \right )}}{C_{1} - 2 x^{2} \log {\left (x \right )} + x^{2}} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (\frac {4 \log {\left (x \right )}}{C_{1} + 2 x^{2} \log {\left (x \right )} - x^{2}} \right )}\right ] \]