85.33.80 problem 81

Internal problem ID [22703]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 81
Date solved : Thursday, October 02, 2025 at 09:11:08 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 3 y^{2}+4 y x +\left (x^{2}+2 y x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 71

ode:=3*y(x)^2+4*x*y(x)+(x^2+2*x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-x^{3} c_1^{3}+\sqrt {c_1 x \left (c_1^{5} x^{5}+4\right )}}{2 x^{2} c_1^{3}} \\ y &= \frac {-x^{3} c_1^{3}-\sqrt {c_1 x \left (c_1^{5} x^{5}+4\right )}}{2 x^{2} c_1^{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.438 (sec). Leaf size: 118

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

\begin{align*} y(x)&\to \frac {1}{2} \left (-x-\frac {\sqrt {x^5+4 e^{c_1}}}{x^{3/2}}\right )\\ y(x)&\to \frac {1}{2} \left (-x+\frac {\sqrt {x^5+4 e^{c_1}}}{x^{3/2}}\right )\\ y(x)&\to -\frac {x^{5/2}+\sqrt {x^5}}{2 x^{3/2}}\\ y(x)&\to \frac {\sqrt {x^5}}{2 x^{3/2}}-\frac {x}{2} \end{align*}

✓ Sympy. Time used: 0.969 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*y(x) + (x**2 + 2*x*y(x))*Derivative(y(x), x) + 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x \left (\sqrt {\frac {C_{1}}{x^{5}} + 1} - 1\right )}{2}, \ y{\left (x \right )} = \frac {x \left (- \sqrt {\frac {C_{1}}{x^{5}} + 1} - 1\right )}{2}\right ] \]