Internal problem ID [5431]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page
20
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {y-x^{3}+\left (y^{3}+x \right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 20
dsolve((y(x)-x^3)+(x+y(x)^3)*diff(y(x),x)=0,y(x), singsol=all)
\[ -\frac {x^{4}}{4}+x y \relax (x )+\frac {y \relax (x )^{4}}{4}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 75.124 (sec). Leaf size: 1126
DSolve[(y[x]-x^3)+(x+y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}+\sqrt {\frac {6 \sqrt {2} x}{\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}-\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}+\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}} \\ y(x)\to \frac {\sqrt {\frac {6 \sqrt {2} x}{\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}-\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}+\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}-\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}} \\ y(x)\to \frac {\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}-\sqrt {-\frac {6 \sqrt {2} x}{\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}-\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}+\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}} \\ y(x)\to \frac {\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}+\sqrt {-\frac {6 \sqrt {2} x}{\sqrt {\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}-\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}-\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}+\frac {x^4+4 c_1}{\sqrt [3]{3 x^2+\sqrt {9 x^4+\frac {1}{3} \left (x^4+4 c_1\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}} \\ \end{align*}