12.5.10 problem 6

Internal problem ID [1634]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 6
Date solved : Saturday, March 29, 2025 at 11:08:55 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} y^{\prime }-\frac {\left (1+x \right ) y}{3 x}&=y^{4} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 119

ode:=diff(y(x),x)-1/3*(1+x)/x*y(x) = y(x)^4; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {{\left (x \left (c_1 \,{\mathrm e}^{-x}-3 x +3\right )^{2}\right )}^{{1}/{3}}}{c_1 \,{\mathrm e}^{-x}-3 x +3} \\ y &= \frac {{\left (x \left (c_1 \,{\mathrm e}^{-x}-3 x +3\right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{-2 c_1 \,{\mathrm e}^{-x}+6 x -6} \\ y &= \frac {{\left (x \left (c_1 \,{\mathrm e}^{-x}-3 x +3\right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 c_1 \,{\mathrm e}^{-x}-6 x +6} \\ \end{align*}

✓ Mathematica. Time used: 1.862 (sec). Leaf size: 120

ode=D[y[x],x]-(1+x)/(3*x)*y[x]==y[x]^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {e^{x/3} \sqrt [3]{x}}{\sqrt [3]{3 e^x (x-1)-c_1}} \\ y(x)\to \frac {\sqrt [3]{-1} e^{x/3} \sqrt [3]{x}}{\sqrt [3]{3 e^x (x-1)-c_1}} \\ y(x)\to -\frac {(-1)^{2/3} e^{x/3} \sqrt [3]{x}}{\sqrt [3]{3 e^x (x-1)-c_1}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 3.111 (sec). Leaf size: 99

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**4 + Derivative(y(x), x) - (x + 1)*y(x)/(3*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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