82.18.17 problem Ex. 19

Internal problem ID [18768]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 19
Date solved : Monday, March 31, 2025 at 06:09:50 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

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

\begin{align*} y &= c_1 \,x^{2} \\ y &= \sqrt {-3 x^{2}+c_1} \\ y &= -\sqrt {-3 x^{2}+c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.13 (sec). Leaf size: 54

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

\begin{align*} y(x)\to c_1 x^2 \\ y(x)\to -\sqrt {-3 x^2+2 c_1} \\ y(x)\to \sqrt {-3 x^2+2 c_1} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.564 (sec). Leaf size: 32

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

\[ \left [ y{\left (x \right )} = C_{1} x^{2}, \ y{\left (x \right )} = - \sqrt {C_{1} - 3 x^{2}}, \ y{\left (x \right )} = \sqrt {C_{1} - 3 x^{2}}\right ] \]