62.19.1 problem Ex 5

Internal problem ID [12843]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter V, Singular solutions. Article 32. Page 69
Problem number : Ex 5
Date solved : Monday, March 31, 2025 at 07:21:25 AM
CAS classification : [[_homogeneous, `class G`], _Clairaut]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.334 (sec). Leaf size: 49

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

\begin{align*} y(x)\to \frac {4 (x+i c_1)}{c_1{}^2} \\ y(x)\to \frac {4 (x-i c_1)}{c_1{}^2} \\ y(x)\to 0 \\ y(x)\to \frac {1}{x} \\ \end{align*}

✓ Sympy. Time used: 4.082 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = C_{1} \left (- C_{1} x + 2\right ) \]