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 : Monday, January 27, 2025 at 05:15:19 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 119

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]-(1+x)/(3*x)*y[x]==y[x]^4,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*}