60.2.106 problem 682

Internal problem ID [10680]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 682
Date solved : Sunday, March 30, 2025 at 06:20:40 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime }&=\frac {2 a}{x \left (-x y+2 a x y^{2}-8 a^{2}\right )} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 26

ode:=diff(y(x),x) = 2*a/x/(-x*y(x)+2*a*x*y(x)^2-8*a^2); 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {\left (-x y^{2}+4 a \right ) {\mathrm e}^{-4 a y}}{x} = 0 \]

✓ Mathematica. Time used: 0.263 (sec). Leaf size: 39

ode=D[y[x],x] == (2*a)/(x*(-8*a^2 - x*y[x] + 2*a*x*y[x]^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {y(x)^2 e^{-4 a y(x)}}{8 a}-\frac {e^{-4 a y(x)}}{2 x}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.168 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*a/(x*(-8*a**2 + 2*a*x*y(x)**2 - x*y(x))) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \begin {cases} - \frac {y^{2}{\left (x \right )} e^{- 4 a y{\left (x \right )}}}{4 a} & \text {for}\: a \neq 0 \\\frac {y^{3}{\left (x \right )}}{3} - \frac {y^{2}{\left (x \right )}}{4 a} & \text {otherwise} \end {cases} - \frac {e^{- 4 a y{\left (x \right )}}}{x} = 0 \]