30.5.10 problem 10

Internal problem ID [7509]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 04:40:43 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.218 (sec). Leaf size: 145

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

\begin{align*} y &= \frac {\sqrt {\frac {x^{4}-\sqrt {\frac {x^{8}+c_1^{4}}{c_1^{4}}}\, c_1^{2}}{c_1^{2}}}\, c_1}{x} \\ y &= \frac {\sqrt {\frac {x^{4}+\sqrt {\frac {x^{8}+c_1^{4}}{c_1^{4}}}\, c_1^{2}}{c_1^{2}}}\, c_1}{x} \\ y &= -\frac {\sqrt {\frac {x^{4}-\sqrt {\frac {x^{8}+c_1^{4}}{c_1^{4}}}\, c_1^{2}}{c_1^{2}}}\, c_1}{x} \\ y &= -\frac {\sqrt {\frac {x^{4}+\sqrt {\frac {x^{8}+c_1^{4}}{c_1^{4}}}\, c_1^{2}}{c_1^{2}}}\, c_1}{x} \\ \end{align*}

✓ Mathematica. Time used: 12.095 (sec). Leaf size: 233

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

\begin{align*} y(x)&\to -\sqrt {x^2-\frac {\sqrt {x^8-e^{4 c_1}}}{x^2}}\\ y(x)&\to \sqrt {x^2-\frac {\sqrt {x^8-e^{4 c_1}}}{x^2}}\\ y(x)&\to -\sqrt {\frac {x^4+\sqrt {x^8-e^{4 c_1}}}{x^2}}\\ y(x)&\to \sqrt {\frac {x^4+\sqrt {x^8-e^{4 c_1}}}{x^2}}\\ y(x)&\to -\sqrt {x^2-\frac {\sqrt {x^8}}{x^2}}\\ y(x)&\to \sqrt {x^2-\frac {\sqrt {x^8}}{x^2}}\\ y(x)&\to -\sqrt {\frac {\sqrt {x^8}+x^4}{x^2}}\\ y(x)&\to \sqrt {\frac {\sqrt {x^8}+x^4}{x^2}} \end{align*}

✓ Sympy. Time used: 3.593 (sec). Leaf size: 85

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

\[ \left [ y{\left (x \right )} = - \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{8}}}{x^{2}}}, \ y{\left (x \right )} = \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{8}}}{x^{2}}}, \ y{\left (x \right )} = - \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{8}}}{x^{2}}}, \ y{\left (x \right )} = \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{8}}}{x^{2}}}\right ] \]