54.2.100 problem 676

Internal problem ID [11974]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 676
Date solved : Sunday, October 12, 2025 at 02:08:21 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.387 (sec). Leaf size: 53

ode:=diff(y(x),x) = 1/2*(x+1+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3/(1+x); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {3 x^{5}-4 x^{4}+6 x^{3}+12 \ln \left (x +1\right ) x +6 c_{1} x -12 x^{2}-6 \sqrt {4 x^{2} y+1}}{6 x} = 0 \]

✓ Mathematica. Time used: 1.113 (sec). Leaf size: 120

ode=D[y[x],x] == (1/2 + x/2 + x^6*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^8}{16}-\frac {x^7}{6}+\frac {13 x^6}{36}-\frac {5 x^5}{6}-\frac {1}{12} (-11+6 c_1) x^4+\left (-1+\frac {2 c_1}{3}\right ) x^3-\frac {1}{4 x^2}-(-1+c_1) x^2+\left (\frac {x^4}{2}-\frac {2 x^3}{3}+x^2-2 x-2 c_1\right ) \log (x+1)+\log ^2(x+1)+2 c_1 x+c_1{}^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x**6*sqrt(4*x**2*y(x) + 1) + x + 1)/(2*x**3*(x + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out