Internal problem ID [9103]
Internal file name [OUTPUT/8038_Monday_June_06_2022_01_21_18_AM_41971886/index.tex]
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.
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.
\[ \boxed {y^{\prime }+\frac {i \left (16 i x^{2}+16 y^{4}+8 x^{4} y^{2}+x^{8}\right ) x}{32 y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \mathrm {I} x^{9}+8 \,\mathrm {I} y^{2} x^{5}+16 \,\mathrm {I} y^{4} x -16 x^{3}+32 y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\mathrm {I} x^{9}-8 \,\mathrm {I} y^{2} x^{5}-16 \,\mathrm {I} y^{4} x +16 x^{3}}{32 y} \end {array} \]
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) = -((1/16)*I)*x^4*(I*x^6-16)*y(x)-((1/2)*I)*(x^6+2*I)*(diff(y(x), x))/x, y( 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: 0.047 (sec). Leaf size: 233
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 \left (x \right ) &= -\frac {\sqrt {-4 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x^{3} \left (\left (1+i\right ) c_{1} \operatorname {BesselJ}\left (-\frac {2}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\frac {\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} x^{3}}{4}+\left (1+i\right ) \operatorname {BesselY}\left (-\frac {2}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\frac {\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) x^{3}}{4}\right )}}{2 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x} \\ y \left (x \right ) &= \frac {\sqrt {-4 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x^{3} \left (\left (1+i\right ) c_{1} \operatorname {BesselJ}\left (-\frac {2}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\frac {\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} x^{3}}{4}+\left (1+i\right ) \operatorname {BesselY}\left (-\frac {2}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\frac {\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) x^{3}}{4}\right )}}{2 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {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: 39.169 (sec). Leaf size: 836
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 (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right ) \left ((1+i) x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )-\frac {1}{4} \left (x^6+8 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )-\frac {1}{4} c_1 \left (x^6+8 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )}}{x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )} \\ y(x)\to \frac {\sqrt {\left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right ) \left ((1+i) x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )-\frac {1}{4} \left (x^6+8 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )-\frac {1}{4} c_1 \left (x^6+8 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )}}{x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) 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)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right ) \left (-4 i 2^{2/3} \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+4 i 2^{2/3} \operatorname {AiryBiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+\sqrt [6]{-1} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\sqrt [6]{-1} x^4 \operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )}{x^2}}}{2 \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{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)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right ) \left (-4 i 2^{2/3} \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+4 i 2^{2/3} \operatorname {AiryBiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+\sqrt [6]{-1} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\sqrt [6]{-1} x^4 \operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )}{x^2}}}{2 \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )} \\ \end{align*}