60.2.412 problem 990

Internal problem ID [10986]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 990
Date solved : Sunday, March 30, 2025 at 07:37:20 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 44

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

\[ y = \frac {-x^{2} {\mathrm e}^{2 \int F \left (x \right )d x}+c_1 \,x^{2}+{\mathrm e}^{2 \int F \left (x \right )d x}+c_1}{-{\mathrm e}^{2 \int F \left (x \right )d x}+c_1} \]

✓ Mathematica. Time used: 0.261 (sec). Leaf size: 67

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

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x2 F(K[5])dK[5]\right )}{-\int _1^x\exp \left (\int _1^{K[6]}2 F(K[5])dK[5]\right ) F(K[6])dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}

✗ Sympy

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

IndexError : Index out of range: a[1]