Internal problem ID [984]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (x +1\right ) y}{3 x}-y^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 179
dsolve(diff(y(x),x)-(1+x)/(3*x)*y(x)=y(x)^4,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (x \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )^{2}\right )^{\frac {1}{3}}}{{\mathrm e}^{-x} c_{1}-3 x +3} \\ y \relax (x ) = -\frac {\left (x \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )^{2}\right )^{\frac {1}{3}}}{2 \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )}-\frac {i \sqrt {3}\, \left (x \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )^{2}\right )^{\frac {1}{3}}}{2 \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )} \\ y \relax (x ) = -\frac {\left (x \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )^{2}\right )^{\frac {1}{3}}}{2 \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )}+\frac {i \sqrt {3}\, \left (x \left ({\mathrm e}^{-x} c_{1}-3 x +3\right )^{2}\right )^{\frac {1}{3}}}{2 \,{\mathrm e}^{-x} c_{1}-6 x +6} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.271 (sec). Leaf size: 120
DSolve[y'[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*}