54.2.76 problem 652

Internal problem ID [11950]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 652
Date solved : Sunday, October 12, 2025 at 02:05:31 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {2 a +x \sqrt {-y^{2}+4 a x}}{y} \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 27

ode:=diff(y(x),x) = (2*a+x*(-y(x)^2+4*a*x)^(1/2))/y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ -\sqrt {-y^{2}+4 a x}-\frac {x^{2}}{2}-c_1 = 0 \]

✓ Mathematica. Time used: 4.354 (sec). Leaf size: 161

ode=D[y[x],x] == (2*a + x*Sqrt[4*a*x - y[x]^2])/y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {256 a^4 x \left (16 a-x^3\right )+32 a^2 e^{c_1} x^2-e^{2 c_1}}}{32 a^2}\\ y(x)&\to \frac {\sqrt {256 a^4 x \left (16 a-x^3\right )+32 a^2 e^{c_1} x^2-e^{2 c_1}}}{32 a^2}\\ y(x)&\to -\frac {\sqrt {a^4 x \left (16 a-x^3\right )}}{2 a^2}\\ y(x)&\to \frac {\sqrt {a^4 x \left (16 a-x^3\right )}}{2 a^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq((-2*a - x*sqrt(4*a*x - y(x)**2))/y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out