Internal problem ID [9077]
Internal file name [OUTPUT/8012_Monday_June_06_2022_01_16_40_AM_23623157/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 743.
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.
\[ \boxed {y^{\prime }+\frac {i \left (8 i x +16 y^{4}+8 y^{2} x^{2}+x^{4}\right )}{32 y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 16 \,\mathrm {I} y^{4}+8 \,\mathrm {I} y^{2} x^{2}+\mathrm {I} x^{4}-8 x +32 y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-16 \,\mathrm {I} y^{4}-8 \,\mathrm {I} y^{2} x^{2}-\mathrm {I} x^{4}+8 x}{32 y} \end {array} \]
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*(I*x^3-8)*y(x)-((1/2)*I)*x^2*(diff(y(x), x)), y(x)` *** S 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: 0.062 (sec). Leaf size: 264
dsolve(diff(y(x),x) = -1/32*I*(8*I*x+16*y(x)^4+8*x^2*y(x)^2+x^4)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {\left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, -\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, -\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )-\frac {x^{2} \left (\operatorname {AiryAi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )\right )}{2}\right ) \left (\operatorname {AiryAi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )\right )}}{2 \operatorname {AiryAi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right ) c_{1} +2 \operatorname {AiryBi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {\left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, -\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, -\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )-\frac {x^{2} \left (\operatorname {AiryAi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )\right )}{2}\right ) \left (\operatorname {AiryAi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )\right )}}{2 \operatorname {AiryAi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right ) c_{1} +2 \operatorname {AiryBi}\left (-\frac {\left (-i+\sqrt {3}\right ) x}{2}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.963 (sec). Leaf size: 553
DSolve[y'[x] == ((-1/32*I)*((8*I)*x + x^4 + 8*x^2*y[x]^2 + 16*y[x]^4))/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-\left (\left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right ) \left (x^2 \operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )+\left (-2-2 i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )\right )}}{2 \left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )} \\ y(x)\to \frac {\sqrt {-\left (\left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right ) \left (x^2 \operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )+\left (-2-2 i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )\right )}}{2 \left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )} \\ y(x)\to -\frac {\sqrt {-\operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right ) \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )}}{2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )} \\ y(x)\to \frac {\sqrt {-\operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right ) \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )}}{2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )} \\ \end{align*}