50.4.3 problem 3

Internal problem ID [7852]
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
Date solved : Monday, January 27, 2025 at 03:26:30 PM
CAS classification : [_exact, _rational]

\begin{align*} y-x^{3}+\left (x +y^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 20

dsolve((y(x)-x^3)+(x+y(x)^3)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ -\frac {x^{4}}{4}+x y+\frac {y^{4}}{4}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.181 (sec). Leaf size: 1210

DSolve[(y[x]-x^3)+(x+y[x]^3)*D[y[x],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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\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 {\sqrt [3]{3} \left (x^4+4 c_1\right )}{\sqrt [3]{9 x^2+\sqrt {3} \sqrt {27 x^4+\left (x^4+4 c_1\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}} \\ \end{align*}