30.4.10 problem 10

Internal problem ID [7493]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.5, Special Integrating Factors. Exercises. page 69
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 04:39:36 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.323 (sec). Leaf size: 71

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

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

✓ Sympy. Time used: 0.850 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = \frac {- x - \sqrt {C_{1} - 4 x^{4} + x^{2}}}{2 x}, \ y{\left (x \right )} = \frac {- x + \sqrt {C_{1} - 4 x^{4} + x^{2}}}{2 x}\right ] \]