62.8.2 problem Ex 2

Internal problem ID [12751]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 15. Page 22
Problem number : Ex 2
Date solved : Wednesday, March 05, 2025 at 08:24:32 PM
CAS classification : [_separable]

\begin{align*} y^{2} \left (3 y-6 x y^{\prime }\right )-x \left (y-2 x y^{\prime }\right )&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 31

ode:=y(x)^2*(3*y(x)-6*x*diff(y(x),x))-x*(y(x)-2*x*diff(y(x),x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {3}\, \sqrt {x}}{3} \\ y &= \frac {\sqrt {3}\, \sqrt {x}}{3} \\ y &= \sqrt {x}\, c_{1} \\ \end{align*}

✓ Mathematica. Time used: 1.676 (sec). Leaf size: 78

ode=y[x]^2*(3*y[x]-6*x*D[y[x],x])- x*(4*y[x]-2*x*D[y[x],x])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {i \sqrt {x} \sqrt {W\left (-3 e^{-4-3 c_1} x^3\right )}}{\sqrt {3}} \\ y(x)\to \frac {i \sqrt {x} \sqrt {W\left (-3 e^{-4-3 c_1} x^3\right )}}{\sqrt {3}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.298 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(-2*x*Derivative(y(x), x) + y(x)) + (-6*x*Derivative(y(x), x) + 3*y(x))*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \]