8.5.15 problem 15

Internal problem ID [743]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 15
Date solved : Tuesday, March 04, 2025 at 11:39:25 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y \left (3 x +y\right )+x \left (x +y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 59

ode:=y(x)*(3*x+y(x))+x*(x+y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-c_1 \,x^{2}-\sqrt {x^{4} c_1^{2}+1}}{c_1 x} \\ y &= \frac {-c_1 \,x^{2}+\sqrt {x^{4} c_1^{2}+1}}{c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.626 (sec). Leaf size: 93

ode=y[x]*(3*x+y[x])+x*(x+y[x])*D[y[x],x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.227 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + y(x))*Derivative(y(x), x) + (3*x + y(x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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