54.2.133 problem 709

Internal problem ID [12007]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 709
Date solved : Sunday, October 12, 2025 at 02:09:26 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 39

ode:=diff(y(x),x) = (2*a*x+2*a+x^3*(-y(x)^2+4*a*x)^(1/2))/(1+x)/y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ -\sqrt {-y^{2}+4 a x}-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (x +1\right )-c_1 = 0 \]

✓ Mathematica. Time used: 2.293 (sec). Leaf size: 143

ode=D[y[x],x] == (2*a + 2*a*x + x^3*Sqrt[4*a*x - y[x]^2])/((1 + x)*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{6} \sqrt {144 a x-\left (2 x^3-3 x^2+6 x+6 c_1\right ){}^2+12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)-36 \log ^2(x+1)}\\ y(x)&\to \frac {1}{6} \sqrt {144 a x-\left (2 x^3-3 x^2+6 x+6 c_1\right ){}^2+12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)-36 \log ^2(x+1)} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*a*x + 2*a + x**3*sqrt(4*a*x - y(x)**2))/((x + 1)*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out