64.6.2 problem 2

Internal problem ID [13352]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Miscellaneous Review. Exercises page 60
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 05:31:31 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.269 (sec). Leaf size: 1123

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

\begin{align*} y &= -\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {{\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} \left (2 \,2^{{1}/{3}} x^{2}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}+4 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}\right )}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {{\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} \left (2 \,2^{{1}/{3}} x^{2}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}+4 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}\right )}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= -\frac {\sqrt {3}\, \sqrt {\left (i \left (-2 \,2^{{1}/{3}} x^{2}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}\right ) \sqrt {3}-2 \,2^{{1}/{3}} x^{2}-2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= \frac {\sqrt {3}\, \sqrt {\left (i \left (-2 \,2^{{1}/{3}} x^{2}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}\right ) \sqrt {3}-2 \,2^{{1}/{3}} x^{2}-2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= -\frac {\sqrt {3}\, \sqrt {\left (-i \left (-2 \,2^{{1}/{3}} x^{2}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}\right ) \sqrt {3}-2 \,2^{{1}/{3}} x^{2}-2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= \frac {\sqrt {3}\, \sqrt {\left (-i \left (-2 \,2^{{1}/{3}} x^{2}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}\right ) \sqrt {3}-2 \,2^{{1}/{3}} x^{2}-2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{{1}/{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{{1}/{3}} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 34.443 (sec). Leaf size: 356

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

\begin{align*} y(x)\to \frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4\right ){}^{2/3}}{6^{2/3} x^2 \sqrt [3]{\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4}} \\ y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (2 \sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+18 c_1 x^4\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3}{12 x^2 \sqrt [3]{\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4}} \\ y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+18 c_1 x^4\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3}{12 x^2 \sqrt [3]{\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4}} \\ \end{align*}