30.3.10 problem 10

Internal problem ID [7466]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.4, Exact equations. Exercises. page 64
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 04:36:32 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.269 (sec). Leaf size: 102

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

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

✓ Sympy. Time used: 0.826 (sec). Leaf size: 34

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

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