Internal problem ID [8349]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 769.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {i \left (16 i x^{2}+16 y^{4}+8 x^{4} y^{2}+x^{8}\right ) x}{32 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 251
dsolve(diff(y(x),x) = -1/32*I*(16*I*x^2+16*y(x)^4+8*x^4*y(x)^2+x^8)*x/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {4}\, \sqrt {\left (\BesselJ \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) \left (-2 \left (\frac {x^{6}}{8}+i\right ) c_{1} \BesselJ \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (-\frac {x^{6}}{4}-2 i\right ) \BesselY \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (1+i\right ) \left (\BesselJ \left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1}+\BesselY \left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x^{3}\right )}}{2 \left (\BesselJ \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x} \\ y \relax (x ) = \frac {\sqrt {4}\, \sqrt {\left (\BesselJ \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) \left (-2 \left (\frac {x^{6}}{8}+i\right ) c_{1} \BesselJ \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (-\frac {x^{6}}{4}-2 i\right ) \BesselY \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (1+i\right ) \left (\BesselJ \left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1}+\BesselY \left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x^{3}\right )}}{2 \left (\BesselJ \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1}+\BesselY \left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 40.565 (sec). Leaf size: 557
DSolve[y'[x] == ((-1/32*I)*x*((16*I)*x^2 + x^8 + 8*x^4*y[x]^2 + 16*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 {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right ) \left ((-4-4 i) x^3 Y_{\frac {4}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+\left (x^6+8 i\right ) Y_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \left (8 i J_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )-x^6 J_{\frac {7}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )\right )\right )}}{2 x \left (Y_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )} \\ y(x)\to \frac {\sqrt {-\left (\left (Y_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right ) \left ((-4-4 i) x^3 Y_{\frac {4}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+\left (x^6+8 i\right ) Y_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \left (8 i J_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )-x^6 J_{\frac {7}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )\right )\right )}}{2 x \left (Y_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 J_{\frac {1}{3}}\left (\left (\frac {1}{3}-\frac {i}{3}\right ) 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 {i x^6}{18}\right ) \left (x^6 \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {i x^6}{18}\right )+24 i \, _0\tilde {F}_1\left (;\frac {1}{3};\frac {i x^6}{18}\right )\right )}}{2 x \sqrt [3]{(1-i) x^3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {i x^6}{18}\right )} \\ y(x)\to -\frac {\sqrt {-\left ((1-i) x^3\right )^{2/3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {i x^6}{18}\right ) \left (x^6 \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {i x^6}{18}\right )+24 i \, _0\tilde {F}_1\left (;\frac {1}{3};\frac {i x^6}{18}\right )\right )}}{2 x \sqrt [3]{(1-i) x^3} \, _0\tilde {F}_1\left (;\frac {4}{3};\frac {i x^6}{18}\right )} \\ \end{align*}