75.11.1 problem 260

Internal problem ID [16781]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 260
Date solved : Monday, March 31, 2025 at 03:17:11 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.139 (sec). Leaf size: 71

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

\begin{align*} y &= -2 \sqrt {x +1} \\ y &= 2 \sqrt {x +1} \\ y &= \sqrt {-c_1^{2}+2 c_1 x -x^{2}+4 x +4} \\ y &= -\sqrt {-x^{2}+\left (2 c_1 +4\right ) x -c_1^{2}+4} \\ \end{align*}

✓ Mathematica. Time used: 0.472 (sec). Leaf size: 65

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

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {-4 x^2-4 (-4+c_1) x+16-c_1{}^2} \\ y(x)\to \frac {1}{2} \sqrt {-4 x^2-4 (-4+c_1) x+16-c_1{}^2} \\ \end{align*}

✗ Sympy

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

Timed Out