54.1.478 problem 491

Internal problem ID [11792]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 491
Date solved : Tuesday, September 30, 2025 at 10:37:09 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+a \,x^{2}+\left (a -1\right ) b&=0 \end{align*}

✓ Maple. Time used: 0.394 (sec). Leaf size: 88

ode:=y(x)^2*diff(y(x),x)^2+2*a*x*y(x)*diff(y(x),x)+(-a+1)*y(x)^2+x^2*a+(a-1)*b = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {-a \,x^{2}+b} \\ y &= -\sqrt {-a \,x^{2}+b} \\ y &= \sqrt {c_1^{2} a -2 c_1 a x -c_1^{2}+2 c_1 x -x^{2}+b} \\ y &= -\sqrt {\left (a -1\right ) c_1^{2}-2 x \left (a -1\right ) c_1 -x^{2}+b} \\ \end{align*}

✓ Mathematica. Time used: 0.751 (sec). Leaf size: 65

ode=(-1 + a)*b + a*x^2 + (1 - a)*y[x]^2 + 2*a*x*y[x]*D[y[x],x] + y[x]^2*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

Timed Out