54.2.279 problem 857

Internal problem ID [12153]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 857
Date solved : Sunday, October 12, 2025 at 02:24:11 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y} \end{align*}

✓ Maple. Time used: 0.192 (sec). Leaf size: 32

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

\[ c_{1} +x^{4}+\frac {4 x^{3}}{3}+4 x -\sqrt {x^{2}-2 x +1+8 y} = 0 \]

✓ Mathematica. Time used: 0.495 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to \frac {x^8}{8}+\frac {x^7}{3}+\frac {2 x^6}{9}+x^5+\left (\frac {4}{3}-c_1\right ) x^4-\frac {4 c_1 x^3}{3}+\frac {15 x^2}{8}+\left (\frac {1}{4}-4 c_1\right ) x-\frac {1}{8}+2 c_1{}^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*sqrt(x**2 - 2*x + 8*y(x) + 1) - x**2*sqrt(x**2 - 2*x + 8*y(x) + 1) + x/4 - sqrt(x**2 - 2*x + 8*y(x) + 1) + Derivative(y(x), x) - 1/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out