32.6.47 problem Exercise 12.47, page 103

Internal problem ID [5912]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.47, page 103
Date solved : Monday, January 27, 2025 at 01:27:01 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 2 y^{3} y^{\prime }+x y^{2}-x^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.361 (sec). Leaf size: 641

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

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

✓ Solution by Mathematica

Time used: 60.145 (sec). Leaf size: 714

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

\begin{align*} y(x)\to -\frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}} \\ y(x)\to -\frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\ y(x)\to \frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\ y(x)\to -\frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\ y(x)\to \frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\ \end{align*}