Internal problem ID [4490]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 2, First order differential equations. Review problems. page 79
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (x^{2}-\frac {2}{y^{3}}\right ) y^{\prime }+2 y x -3 x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 878
dsolve((x^2-2*y(x)^(-3))*diff(y(x),x)+(2*x*y(x)-3*x^2)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}{6 x^{2}}+\frac {2 \left (-x^{3}+c_{1}\right )^{2}}{3 x^{2} \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}-\frac {-x^{3}+c_{1}}{3 x^{2}} \\ y \relax (x ) = -\frac {\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}{12 x^{2}}-\frac {\left (-x^{3}+c_{1}\right )^{2}}{3 x^{2} \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}-\frac {-x^{3}+c_{1}}{3 x^{2}}-\frac {i \sqrt {3}\, \left (\frac {\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}{6 x^{2}}-\frac {2 \left (-x^{3}+c_{1}\right )^{2}}{3 x^{2} \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}{12 x^{2}}-\frac {\left (-x^{3}+c_{1}\right )^{2}}{3 x^{2} \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}-\frac {-x^{3}+c_{1}}{3 x^{2}}+\frac {i \sqrt {3}\, \left (\frac {\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}{6 x^{2}}-\frac {2 \left (-x^{3}+c_{1}\right )^{2}}{3 x^{2} \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 25.991 (sec). Leaf size: 676
DSolve[(x^2-2*y[x]^(-3))*y'[x]+(2*x*y[x]-3*x^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \left (x^3+c_1\right )+\frac {2 \left (x^3+c_1\right ){}^2}{\sqrt [3]{x^9+3 c_1 x^6-\frac {27 x^4}{2}+3 c_1{}^2 x^3+\frac {3}{2} \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+c_1{}^3}}+2^{2/3} \sqrt [3]{2 x^9+6 c_1 x^6-27 x^4+6 c_1{}^2 x^3+3 \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+2 c_1{}^3}}{6 x^2} \\ y(x)\to \frac {4 \left (x^3+c_1\right )-\frac {2 i \left (\sqrt {3}-i\right ) \left (x^3+c_1\right ){}^2}{\sqrt [3]{x^9+3 c_1 x^6-\frac {27 x^4}{2}+3 c_1{}^2 x^3+\frac {3}{2} \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^9+6 c_1 x^6-27 x^4+6 c_1{}^2 x^3+3 \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+2 c_1{}^3}}{12 x^2} \\ y(x)\to \frac {4 \left (x^3+c_1\right )+\frac {2 i \left (\sqrt {3}+i\right ) \left (x^3+c_1\right ){}^2}{\sqrt [3]{x^9+3 c_1 x^6-\frac {27 x^4}{2}+3 c_1{}^2 x^3+\frac {3}{2} \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^9+6 c_1 x^6-27 x^4+6 c_1{}^2 x^3+3 \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+2 c_1{}^3}}{12 x^2} \\ y(x)\to 0 \\ \end{align*}