60.1.439 problem 451

Internal problem ID [10453]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 451
Date solved : Sunday, March 30, 2025 at 04:45:48 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

\begin{align*} \left (x^{2}+a \right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}+b&=0 \end{align*}

✓ Maple. Time used: 0.172 (sec). Leaf size: 78

ode:=(x^2+a)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2+b = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {-a b \left (x^{2}+a \right )}}{a} \\ y &= -\frac {\sqrt {-a b \left (x^{2}+a \right )}}{a} \\ y &= c_1 x -\sqrt {-c_1^{2} a -b} \\ y &= c_1 x +\sqrt {-c_1^{2} a -b} \\ \end{align*}

✓ Mathematica. Time used: 0.303 (sec). Leaf size: 96

ode=b + y[x]^2 - 2*x*y[x]*D[y[x],x] + (a + x^2)*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 x-\sqrt {-b-a c_1{}^2} \\ y(x)\to \sqrt {-b-a c_1{}^2}+c_1 x \\ y(x)\to -\frac {\sqrt {-b \left (a+x^2\right )}}{\sqrt {a}} \\ y(x)\to \frac {\sqrt {-b \left (a+x^2\right )}}{\sqrt {a}} \\ \end{align*}

✗ Sympy

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

Timed Out