77.1.47 problem 66 (page 109)

Internal problem ID [17866]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 66 (page 109)
Date solved : Monday, March 31, 2025 at 04:37:38 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.837 (sec). Leaf size: 56

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \frac {x \left (c_1^{2} {\mathrm e}^{-x}-{\mathrm e}^{x}\right )}{2 c_1} \\ y &= -\frac {x \left (-c_1^{2} {\mathrm e}^{x}+{\mathrm e}^{-x}\right )}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 26

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

\begin{align*} y(x)\to -x \sinh (x-c_1) \\ y(x)\to x \sinh (x+c_1) \\ \end{align*}

✗ Sympy

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

Timed Out