79.3.1 problem (a)

Internal problem ID [21091]
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 VI at page 33
Problem number : (a)
Date solved : Thursday, October 02, 2025 at 07:07:24 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=f \left (x \right ) y \ln \left (\frac {1}{y}\right ) \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 17

ode:=diff(y(x),x) = f(x)*y(x)*ln(1/y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {{\mathrm e}^{-\int f \left (x \right )d x}}{c_1}} \]

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to \exp \left (-\exp \left (-\int _1^xf(K[1])dK[1]-c_1\right )\right )\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.248 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = e^{C_{1} e^{- \int f{\left (x \right )}\, dx}} \]