28.1.57 problem 58

Internal problem ID [4363]
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
Date solved : Monday, January 27, 2025 at 09:09:13 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 16

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

\[ -c_{1} y \left (x \right )+x -\frac {y \left (x \right )^{4}}{3} = 0 \]

✓ Solution by Mathematica

Time used: 0.108 (sec). Leaf size: 997

DSolve[(y[x]^3+x/y[x])*D[y[x],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*}