Internal problem ID [9080]
Internal file name [OUTPUT/8015_Monday_June_06_2022_01_17_03_AM_34533863/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 746.
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.
\[ \boxed {y^{\prime }+\frac {i \left (i x +x^{4}+2 x^{2} y^{2}+y^{4}\right )}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \mathrm {I} y^{4}+2 \,\mathrm {I} y^{2} x^{2}+\mathrm {I} x^{4}-x +y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\mathrm {I} y^{4}-2 \,\mathrm {I} y^{2} x^{2}-\mathrm {I} x^{4}+x}{y} \end {array} \]
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*I)*x*(I*x^3-1)*y(x)-(4*I)*x^2*(diff(y(x), x)), y(x)` *** Sublevel 2 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: 232
dsolve(diff(y(x),x) = -I*(I*x+x^4+2*x^2*y(x)^2+y(x)^4)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {\left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )-2 x^{2} \left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right )\right )}}{2 \operatorname {AiryAi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1} +2 \operatorname {AiryBi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right )} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {\left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )-2 x^{2} \left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right )\right )}}{2 \operatorname {AiryAi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1} +2 \operatorname {AiryBi}\left (-\left (-8 i\right )^{\frac {1}{3}} x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.529 (sec). Leaf size: 413
DSolve[y'[x] == ((-I)*(I*x + x^4 + 2*x^2*y[x]^2 + y[x]^4))/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right ) \left (-2 x^2 \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) c_1 \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )} \\ y(x)\to \frac {\sqrt {\left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right ) \left (-2 x^2 \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) c_1 \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )} \\ y(x)\to -\frac {\sqrt {-\operatorname {AiryAi}\left (2 (-1)^{5/6} x\right ) \left (2 x^2 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )+\left (-1-i \sqrt {3}\right ) \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )} \\ y(x)\to \frac {\sqrt {-\operatorname {AiryAi}\left (2 (-1)^{5/6} x\right ) \left (2 x^2 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )+\left (-1-i \sqrt {3}\right ) \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )} \\ \end{align*}