54.2.159 problem 736

Internal problem ID [12033]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 736
Date solved : Wednesday, October 01, 2025 at 12:01:18 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 43

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

\[ y = \frac {c_1 \left (x^{4}+2 x^{3}-x^{2}-2 x -2\right )+x^{2}+1}{1+\left (x^{2}+2 x \right ) c_1} \]

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 39

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

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

✓ Sympy. Time used: 0.312 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = \frac {C_{1} x^{2} + C_{1} + x^{4} + 2 x^{3} - 2 x^{2} - 2 x - 3}{C_{1} + x^{2} + 2 x - 1} \]