78.4.3 problem 3

Internal problem ID [18126]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:30:04 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 20

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

\[ -\frac {x^{4}}{4}+x y+\frac {y^{4}}{4}+c_1 = 0 \]

✓ Solution by Mathematica

Time used: 60.162 (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*}