79.2.4 problem (e)

Internal problem ID [21086]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises XIII at page 24
Problem number : (e)
Date solved : Thursday, October 02, 2025 at 07:07:04 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&={\mathrm e}^{{\mathrm e}} \\ \end{align*}

✓ Maple. Time used: 0.254 (sec). Leaf size: 16

ode:=diff(y(x),x) = y(x)*ln(y(x))/sin(x); 
ic:=[y(1/2*Pi) = exp(exp(1))]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-{\mathrm e} \left (\cot \left (x \right )-\csc \left (x \right )\right )} \]

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 16

ode=D[y[x],x]== y[x]*Log[y[x]]/Sin[x]; 
ic={y[Pi/2]==Exp[Exp[1]]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{e^{1-\text {arctanh}(\cos (x))}} \end{align*}

✓ Sympy. Time used: 0.311 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = e^{- \frac {e i \sqrt {\cos {\left (x \right )} - 1}}{\sqrt {\cos {\left (x \right )} + 1}}} \]