55.1.6 problem 1.1.6

Internal problem ID [13226]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, First-Order differential equations
Problem number : 1.1.6
Date solved : Wednesday, October 01, 2025 at 03:37:19 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\[ y = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {1}{-f \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 33

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{K[1]-f(K[1])}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.465 (sec). Leaf size: 24

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

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