78.8.36 problem 36

Internal problem ID [18234]
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. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 36
Date solved : Tuesday, January 28, 2025 at 11:42:59 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} 3 x^{2} y-y^{3}-\left (3 x y^{2}-x^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 244

dsolve((3*x^2*y(x)-y(x)^3) - (3*x*y(x)^2-x^3) * diff(y(x),x)= 0,y(x), singsol=all)
 

\begin{align*} y &= \frac {12^{{1}/{3}} \left (x^{4} c_{1}^{2} 12^{{1}/{3}}+{\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{2}/{3}}\right )}{6 c_{1} x {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{1}/{3}}} \\ y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-1-i \sqrt {3}\right ) {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{2}/{3}}+c_{1}^{2} 2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{4}\right )}{12 {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{1}/{3}} c_{1} x} \\ y &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_{1}^{2} 2^{{2}/{3}} x^{4}\right ) 3^{{1}/{3}}}{12 {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{1}/{3}} c_{1} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.220 (sec). Leaf size: 338

DSolve[(3*x^2*y[x]-y[x]^3) - (3*x*y[x]^2-x^3) * D[y[x],x]== 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\frac {\sqrt {-12 x^8+81 e^{2 c_1}}-9 e^{c_1}}{x}\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\frac {\sqrt {-12 x^8+81 e^{2 c_1}}-9 e^{c_1}}{x}}} \\ y(x)\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\frac {\sqrt {-12 x^8+81 e^{2 c_1}}-9 e^{c_1}}{x}\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^2}{12 \sqrt [3]{\frac {\sqrt {-12 x^8+81 e^{2 c_1}}-9 e^{c_1}}{x}}} \\ y(x)\to -\frac {i \left (2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (\frac {\sqrt {-12 x^8+81 e^{2 c_1}}-9 e^{c_1}}{x}\right ){}^{2/3}+\sqrt [3]{2} \sqrt [6]{3} \left (-6-2 i \sqrt {3}\right ) x^2\right )}{12 \sqrt [3]{\frac {\sqrt {-12 x^8+81 e^{2 c_1}}-9 e^{c_1}}{x}}} \\ \end{align*}