Internal problem ID [1958]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 8, page 34
Problem number: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _exact, _rational]
\[ \boxed {\frac {2 y}{x^{3}}+\frac {2 x}{y^{2}}-\left (\frac {1}{x^{2}}+\frac {2 x^{2}}{y^{3}}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.625 (sec). Leaf size: 231
dsolve(2*(y(x)/x^3+x/y(x)^2)=(1/x^2+2*x^2/y(x)^3)*diff(y(x),x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{\frac {4}{3}} \operatorname {RootOf}\left (x^{14} \textit {\_Z}^{18}+6 x^{\frac {38}{3}} \textit {\_Z}^{16}+15 x^{\frac {34}{3}} \textit {\_Z}^{14}+\left (20 x^{10}-27 c_{1} x^{8}\right ) \textit {\_Z}^{12}+\left (15 x^{\frac {26}{3}}-81 x^{\frac {20}{3}} c_{1} \right ) \textit {\_Z}^{10}+\left (6 x^{\frac {22}{3}}-108 x^{\frac {16}{3}} c_{1} \right ) \textit {\_Z}^{8}+\left (x^{6}-81 c_{1} x^{4}\right ) \textit {\_Z}^{6}-36 x^{\frac {8}{3}} c_{1} \textit {\_Z}^{4}-9 x^{\frac {4}{3}} c_{1} \textit {\_Z}^{2}-c_{1} \right )^{2}+1}{\operatorname {RootOf}\left (x^{14} \textit {\_Z}^{18}+6 x^{\frac {38}{3}} \textit {\_Z}^{16}+15 x^{\frac {34}{3}} \textit {\_Z}^{14}+\left (20 x^{10}-27 c_{1} x^{8}\right ) \textit {\_Z}^{12}+\left (15 x^{\frac {26}{3}}-81 x^{\frac {20}{3}} c_{1} \right ) \textit {\_Z}^{10}+\left (6 x^{\frac {22}{3}}-108 x^{\frac {16}{3}} c_{1} \right ) \textit {\_Z}^{8}+\left (x^{6}-81 c_{1} x^{4}\right ) \textit {\_Z}^{6}-36 x^{\frac {8}{3}} c_{1} \textit {\_Z}^{4}-9 x^{\frac {4}{3}} c_{1} \textit {\_Z}^{2}-c_{1} \right )^{2}} \]
✓ Solution by Mathematica
Time used: 10.8 (sec). Leaf size: 414
DSolve[2*(y[x]/x^3+x/y[x]^2)==(1/x^2+2*x^2/y[x]^3)*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (c_1 x^2+\frac {c_1{}^2 x^4}{\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}}+\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}\right ) \\ y(x)\to \frac {1}{12} \left (4 c_1 x^2-\frac {2 \left (1+i \sqrt {3}\right ) c_1{}^2 x^4}{\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 c_1{}^3 x^6+27 x^4+3 \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}\right ) \\ y(x)\to \frac {1}{12} \left (4 c_1 x^2+\frac {2 i \left (\sqrt {3}+i\right ) c_1{}^2 x^4}{\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 c_1{}^3 x^6+27 x^4+3 \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}\right ) \\ \end{align*}