60.1.123 problem 126

Internal problem ID [10137]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 126
Date solved : Sunday, March 30, 2025 at 03:19:32 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x y^{\prime }-y f \left (x y\right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

ode:=x*diff(y(x),x)-y(x)*f(x*y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.203 (sec). Leaf size: 115

ode=x*D[y[x],x] - y[x]*f[x*y[x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(-f(x K[2])-1) K[2]}-\int _1^x\left (\frac {f''(K[1] K[2])}{f(K[1] K[2])+1}-\frac {f(K[1] K[2]) f''(K[1] K[2])}{(f(K[1] K[2])+1)^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {f(K[1] y(x))}{(f(K[1] y(x))+1) K[1]}dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.186 (sec). Leaf size: 17

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

\[ \int \limits ^{x y{\left (x \right )}} \frac {1}{y \left (f{\left (y \right )} + 1\right )}\, dy = C_{1} + \log {\left (x \right )} \]