54.1.350 problem 357

Internal problem ID [11664]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 357
Date solved : Sunday, October 12, 2025 at 01:50:49 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right )&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 13

ode:=x*diff(y(x),x)*ln(x)*sin(y(x))+cos(y(x))*(1-x*cos(y(x))) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.654 (sec). Leaf size: 53

ode=Cos[y[x]]*(1 - x*Cos[y[x]]) + x*Log[x]*Sin[y[x]]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.560 (sec). Leaf size: 26

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

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