Internal problem ID [9900]
Internal file name [OUTPUT/8847_Monday_June_06_2022_05_37_30_AM_45443521/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1578.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+y a^{4}-\lambda \left (a x -b \right ) \left (y^{\prime \prime }-a^{2} y\right )=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }-2 a^{2} \left (\frac {d}{d x}y^{\prime }\right )+y a^{4}-\lambda \left (a x -b \right ) \left (\frac {d}{d x}y^{\prime }-a^{2} y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 4; missing the dependent variable trying a solution in terms of MeijerG functions trying reduction of order using simple exponentials -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = _b(_a)*_a*a*lambda-_b(_a)*b*lambda+2*(diff(_b(_a), _a))*a, _b(_a)` 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 <- reduction of order using simple exponentials successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 89
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)-2*a^2*diff(diff(y(x),x),x)+a^4*y(x)-lambda*(a*x-b)*(diff(diff(y(x),x),x)-a^2*y(x))=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{a x} \left (\int {\mathrm e}^{-2 a x} \left (\int {\mathrm e}^{a x} \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (-a \lambda \right )^{\frac {1}{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{a \lambda }\right )+c_{4} \operatorname {AiryBi}\left (-\frac {\left (-a \lambda \right )^{\frac {1}{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{a \lambda }\right )\right )d x +c_{2} \right )d x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 40.473 (sec). Leaf size: 130
DSolve[a^4*y[x] - 2*a^2*y''[x] - \[Lambda]*(-b + a*x)*(-(a^2*y[x]) + y''[x]) + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} \left (c_3 \int _1^x2 a e^{2 a K[1]} \int e^{-a K[1]} \operatorname {AiryAi}\left (\frac {a^2+\lambda K[1] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[1]dK[1]+c_4 \int _1^x2 a e^{2 a K[2]} \int e^{-a K[2]} \operatorname {AiryBi}\left (\frac {a^2+\lambda K[2] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[2]dK[2]+c_2 e^{2 a x}+c_1\right ) \]