54.2.104 problem 680

Internal problem ID [11978]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 680
Date solved : Tuesday, September 30, 2025 at 11:51:16 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.110 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.525 (sec). Leaf size: 39

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

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

✓ Sympy. Time used: 0.972 (sec). Leaf size: 22

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

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