54.2.82 problem 658

Internal problem ID [11956]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 658
Date solved : Tuesday, September 30, 2025 at 11:47:54 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.172 (sec). Leaf size: 28

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

\[ c_{1} +4 \ln \left (x +1\right )-\frac {1}{4}-\sqrt {x^{2}-2 x +1+8 y} = 0 \]

✓ Mathematica. Time used: 0.888 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to \frac {1}{8} \left (-x^2+2 x-1+16 c_1{}^2\right )+2 \log ^2\left (\frac {1}{x+1}\right )+4 c_1 \log \left (\frac {1}{x+1}\right ) \end{align*}

✓ Sympy. Time used: 0.969 (sec). Leaf size: 24

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

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