Internal problem ID [8339]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 759.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {i \left (54 i x^{2}+81 y^{4}+18 x^{4} y^{2}+x^{8}\right ) x}{243 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.164 (sec). Leaf size: 305
dsolve(diff(y(x),x) = -1/243*I*(54*I*x^2+81*y(x)^4+18*x^4*y(x)^2+x^8)*x/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {3}\, \sqrt {\left (\BesselJ \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \left (-9 \left (\frac {x^{6}}{27}+i\right ) c_{1} \BesselJ \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )+\left (-\frac {x^{6}}{3}-9 i\right ) \BesselY \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )+\left (1+i\right ) \sqrt {6}\, \left (c_{1} \BesselJ \left (\frac {4}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )+\BesselY \left (\frac {4}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x^{3}\right )}}{3 \left (\BesselJ \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x} \\ y \relax (x ) = \frac {\sqrt {3}\, \sqrt {\left (\BesselJ \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \left (-9 \left (\frac {x^{6}}{27}+i\right ) c_{1} \BesselJ \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )+\left (-\frac {x^{6}}{3}-9 i\right ) \BesselY \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )+\left (1+i\right ) \sqrt {6}\, \left (c_{1} \BesselJ \left (\frac {4}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )+\BesselY \left (\frac {4}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x^{3}\right )}}{3 \left (\BesselJ \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 38.212 (sec). Leaf size: 781
DSolve[y'[x] == ((-1/243*I)*x*((54*I)*x^2 + x^8 + 18*x^4*y[x]^2 + 81*y[x]^4))/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-\left (\left (Y_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right ) \left ((-3-3 i) \sqrt {6} x^3 Y_{\frac {4}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+\left (x^6+27 i\right ) Y_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \left (27 i J_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )-x^6 J_{\frac {7}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )\right )\right )}}{3 x \left (Y_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )} \\ y(x)\to \frac {\sqrt {-\left (\left (Y_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right ) \left ((-3-3 i) \sqrt {6} x^3 Y_{\frac {4}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+\left (x^6+27 i\right ) Y_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \left (27 i J_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )-x^6 J_{\frac {7}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )\right )\right )}}{3 x \left (Y_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )} \\ y(x)\to \frac {\sqrt {-\left ((1-i) x^3\right )^{2/3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right ) \left (x^6 \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right )+81 i \, _0\tilde {F}_1\left (;\frac {1}{3};\frac {4 i x^6}{243}\right )\right )}}{3 x \sqrt [3]{(1-i) x^3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right )} \\ y(x)\to -\frac {\sqrt {-\left ((1-i) x^3\right )^{2/3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right ) \left (x^6 \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right )+81 i \, _0\tilde {F}_1\left (;\frac {1}{3};\frac {4 i x^6}{243}\right )\right )}}{3 x \sqrt [3]{(1-i) x^3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right )} \\ y(x)\to \frac {\sqrt {-\left ((1-i) x^3\right )^{2/3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right ) \left (x^6 \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right )+81 i \, _0\tilde {F}_1\left (;\frac {1}{3};\frac {4 i x^6}{243}\right )\right )}}{3 x \sqrt [3]{(1-i) x^3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {4 i x^6}{243}\right )} \\ \end{align*}