54.2.140 problem 717

Internal problem ID [12014]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 717
Date solved : Tuesday, September 30, 2025 at 11:55:39 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.298 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.473 (sec). Leaf size: 44

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

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

✓ Sympy. Time used: 1.480 (sec). Leaf size: 26

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

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