Internal problem ID [2693]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 58.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {\left (y^{3}+\frac {x}{y}\right ) y^{\prime }-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 16
dsolve((y(x)^3+x/y(x))*diff(y(x),x)=1,y(x), singsol=all)
\[ -y \relax (x ) c_{1}+x -\frac {y \relax (x )^{4}}{3} = 0 \]
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 997
DSolve[(y[x]^3+x/y[x])*y'[x]==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}+\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}}-\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}} \\ \end{align*}