8.5.32 problem 32

Internal problem ID [760]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 32
Date solved : Saturday, March 29, 2025 at 10:20:10 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 4 x -y+\left (-x +6 y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 51

ode:=4*x-y(x)+(-x+6*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c_1 x -\sqrt {-23 x^{2} c_1^{2}+12}}{6 c_1} \\ y &= \frac {c_1 x +\sqrt {-23 x^{2} c_1^{2}+12}}{6 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.392 (sec). Leaf size: 106

ode=4*x-y[x]+(-x+6*y[x])*D[y[x],x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (x-\sqrt {-23 x^2+12 e^{c_1}}\right ) \\ y(x)\to \frac {1}{6} \left (x+\sqrt {-23 x^2+12 e^{c_1}}\right ) \\ y(x)\to \frac {1}{6} \left (x-\sqrt {23} \sqrt {-x^2}\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {23} \sqrt {-x^2}+x\right ) \\ \end{align*}

✓ Sympy. Time used: 1.248 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x + (-x + 6*y(x))*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x}{6} - \frac {\sqrt {C_{1} - 23 x^{2}}}{6}, \ y{\left (x \right )} = \frac {x}{6} + \frac {\sqrt {C_{1} - 23 x^{2}}}{6}\right ] \]