77.1.94 problem 121 (page 179)

Internal problem ID [17913]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 121 (page 179)
Date solved : Monday, March 31, 2025 at 04:50:01 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x \left (x y+1\right ) y^{\prime \prime }+x^{2} {y^{\prime }}^{2}+\left (4 x y+2\right ) y^{\prime }+y^{2}+1&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 53

ode:=x*(1+x*y(x))*diff(diff(y(x),x),x)+x^2*diff(y(x),x)^2+(4*x*y(x)+2)*diff(y(x),x)+y(x)^2+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-1-\sqrt {-2 c_2 x -x^{2}+2 c_1 +1}}{x} \\ y &= \frac {-1+\sqrt {-2 c_2 x -x^{2}+2 c_1 +1}}{x} \\ \end{align*}

✓ Mathematica. Time used: 1.523 (sec). Leaf size: 84

ode=x*(x*y[x]+1)*D[y[x],{x,2}]+x^2*D[y[x],x]^2+(4*x*y[x]+2)*D[y[x],x]+y[x]^2+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1+\sqrt {\frac {1}{x^2}} \sqrt {-x^2 \left (x^2-c_2 x-1-2 c_1\right )}}{x} \\ y(x)\to \frac {-1+\sqrt {\frac {1}{x^2}} \sqrt {-x^2 \left (x^2-c_2 x-1-2 c_1\right )}}{x} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*x*y(x) + sqrt(-x**4*y(x)*Derivative(y(x), (x, 2)) - x**3*Derivative(y(x), (x, 2)) + 3*x**2*y(x)**2 - x**2 + 4*x*y(x) + 1) - 1)/x**2 cannot be solved by the factorable group method