54.2.236 problem 813

Internal problem ID [12110]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 813
Date solved : Sunday, October 12, 2025 at 02:19:06 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=\frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2} \end{align*}

✓ Maple. Time used: 0.571 (sec). Leaf size: 40

ode:=diff(y(x),x) = 1/2*(-a^(1/2)*x^3+2*(a*x^4+8*y(x))^(1/2)+2*x^2*(a*x^4+8*y(x))^(1/2)+2*x^3*(a*x^4+8*y(x))^(1/2))*a^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\sqrt {a \,x^{4}+8 y}}{4}+\frac {\left (-3 x^{4}-4 x^{3}-12 x \right ) \sqrt {a}}{12}-c_{1} = 0 \]

✓ Mathematica. Time used: 0.567 (sec). Leaf size: 64

ode=D[y[x],x] == (Sqrt[a]*(-(Sqrt[a]*x^3) + 2*Sqrt[a*x^4 + 8*y[x]] + 2*x^2*Sqrt[a*x^4 + 8*y[x]] + 2*x^3*Sqrt[a*x^4 + 8*y[x]]))/2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{72} a \left (9 x^8+24 x^7+16 x^6+72 x^5+(87-72 c_1) x^4-96 c_1 x^3+144 x^2-288 c_1 x+144 c_1{}^2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-sqrt(a)*(-sqrt(a)*x**3 + 2*x**3*sqrt(a*x**4 + 8*y(x)) + 2*x**2*sqrt(a*x**4 + 8*y(x)) + 2*sqrt(a*x**4 + 8*y(x)))/2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out