15.4.1 problem 1

Internal problem ID [2914]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 8, page 34
Problem number : 1
Date solved : Tuesday, March 04, 2025 at 03:29:20 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 51

ode:=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 {3 c_1^{2} x^{2}+2}}{2 c_1} \\ y &= \frac {c_1 x +\sqrt {3 c_1^{2} x^{2}+2}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.531 (sec). Leaf size: 106

ode=(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 {3 x^2-2 e^{2 c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (x+\sqrt {3 x^2-2 e^{2 c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (x-\sqrt {3} \sqrt {x^2}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {3} \sqrt {x^2}+x\right ) \\ \end{align*}

✓ Sympy. Time used: 1.283 (sec). Leaf size: 34

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

\[ \left [ y{\left (x \right )} = \frac {x}{2} - \frac {\sqrt {C_{1} + 3 x^{2}}}{2}, \ y{\left (x \right )} = \frac {x}{2} + \frac {\sqrt {C_{1} + 3 x^{2}}}{2}\right ] \]