Internal problem ID [9093]
Internal file name [OUTPUT/8028_Monday_June_06_2022_01_19_44_AM_59008077/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {i \left (54 i x^{2}+81 y^{4}+18 y^{2} x^{4}+x^{8}\right ) x}{243 y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \mathrm {I} x^{9}+18 \,\mathrm {I} y^{2} x^{5}+81 \,\mathrm {I} y^{4} x -54 x^{3}+243 y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\mathrm {I} x^{9}-18 \,\mathrm {I} y^{2} x^{5}-81 \,\mathrm {I} y^{4} x +54 x^{3}}{243 y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -((4/729)*I)*x^4*(I*x^6-54)*y(x)-((1/27)*I)*(4*x^6+27*I)*(diff(y(x), x))/ Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 283
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 \left (x \right ) &= -\frac {\sqrt {-3 x^{3} \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \left (\frac {\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} x^{3}}{3}+\frac {\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) x^{3}}{3}+\left (1+i\right ) \left (\operatorname {BesselJ}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \sqrt {6}\right )}}{3 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x} \\ y \left (x \right ) &= \frac {\sqrt {-3 x^{3} \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \left (\frac {\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} x^{3}}{3}+\frac {\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) x^{3}}{3}+\left (1+i\right ) \left (\operatorname {BesselJ}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \sqrt {6}\right )}}{3 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {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: 37.777 (sec). Leaf size: 1293
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 (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right ) \left ((1+i) \sqrt {6} x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )-\frac {1}{3} \left (x^6+27 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )-\frac {1}{3} c_1 \left (x^6+27 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )}}{\sqrt {3} x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )} \\ y(x)\to \frac {\sqrt {\left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right ) \left ((1+i) \sqrt {6} x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )-\frac {1}{3} \left (x^6+27 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )-\frac {1}{3} c_1 \left (x^6+27 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )}}{\sqrt {3} x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )} \\ y(x)\to -\frac {(-1)^{5/6} x \sqrt {-\frac {\sqrt [6]{-1} \left ((1-i) x^3\right )^{2/3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right ) \left (-18 i \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+18 i \operatorname {AiryBiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+\sqrt [6]{-1} \sqrt [3]{2} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\sqrt [6]{-1} \sqrt [3]{2} x^4 \operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )}{x^2}}}{3 \sqrt [6]{2} \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )} \\ y(x)\to \frac {(-1)^{5/6} x \sqrt {-\frac {\sqrt [6]{-1} \left ((1-i) x^3\right )^{2/3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right ) \left (-18 i \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+18 i \operatorname {AiryBiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+\sqrt [6]{-1} \sqrt [3]{2} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\sqrt [6]{-1} \sqrt [3]{2} x^4 \operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )}{x^2}}}{3 \sqrt [6]{2} \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )} \\ y(x)\to \frac {(-1)^{5/6} x \sqrt {-\frac {\sqrt [6]{-1} \left ((1-i) x^3\right )^{2/3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right ) \left (-18 i \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+18 i \operatorname {AiryBiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+\sqrt [6]{-1} \sqrt [3]{2} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\sqrt [6]{-1} \sqrt [3]{2} x^4 \operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )}{x^2}}}{3 \sqrt [6]{2} \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )} \\ \end{align*}