54.2.89 problem 665

Internal problem ID [11963]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 665
Date solved : Friday, October 03, 2025 at 02:54:22 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.529 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.433 (sec). Leaf size: 39

ode=D[y[x],x] == -1/2*(Sqrt[a]*(Sqrt[a]*x^3 + Sqrt[a]*x^4 - 2*Sqrt[a*x^4 + 8*y[x]]))/(1 + x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {a x^4}{8}+2 a \log ^2(x+1)-4 a c_1 \log (x+1)+2 a c_1{}^2 \end{align*}

✓ Sympy. Time used: 1.080 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \frac {a \left (- x^{4} + 16 \left (C_{1} + \log {\left (x + 1 \right )}\right )^{2}\right )}{8} \]