85.33.52 problem 52

Internal problem ID [22675]
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 : 52
Date solved : Thursday, October 02, 2025 at 09:05:27 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 0.365 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to \frac {1}{2} \left (-x-\sqrt {x} \sqrt {x+4 e^{c_1}}\right )\\ y(x)&\to \frac {1}{2} \left (-x+\sqrt {x} \sqrt {x+4 e^{c_1}}\right )\\ y(x)&\to 0\\ y(x)&\to -x \end{align*}

✓ Sympy. Time used: 0.756 (sec). Leaf size: 32

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

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