Internal problem ID [9864]
Internal file name [OUTPUT/8809_Monday_June_06_2022_05_32_03_AM_13928657/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1540.
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.
\[ \boxed {y^{\prime \prime \prime \prime }+a \left (b x -1\right ) y^{\prime \prime }+a b y^{\prime }+\lambda y=0} \] 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 }+a \left (b x -1\right ) \left (\frac {d}{d x}y^{\prime }\right )+a b y^{\prime }+\lambda y=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 } \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k} \\ \square & {} & \textrm {Rewrite DE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k} k \,x^{k -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +1} \left (k +1\right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =\max \left (0, 2-m \right )}{\sum }}a_{k} k \left (k -1\right ) x^{k -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =\max \left (0, 2-m \right )+m -2}{\sum }}a_{k +2-m} \left (k +2-m \right ) \left (k +1-m \right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =4}{\sum }}a_{k} k \left (k -1\right ) \left (k -2\right ) \left (k -3\right ) x^{k -4} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +4} \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite DE with series expansions}\hspace {3pt} \\ {} & {} & \moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +4} \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (k +1\right )-a_{k +2} \left (k +2\right ) \left (k +1\right ) a +a b a_{k +1} \left (k +1\right )^{2}+\lambda a_{k}\right ) x^{k}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k +4} \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (k +1\right )-a_{k +2} \left (k +2\right ) \left (k +1\right ) a +a b a_{k +1} \left (k +1\right )^{2}+\lambda a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +4}=-\frac {a b \,k^{2} a_{k +1}+2 a b k a_{k +1}-a \,k^{2} a_{k +2}+a b a_{k +1}-3 a k a_{k +2}-2 a_{k +2} a +\lambda a_{k}}{\left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (k +1\right )}\right ] \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 --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+a*(b*x-1)*diff(diff(y(x),x),x)+a*b*diff(y(x),x)+lambda*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[\[Lambda]*y[x] + a*b*y'[x] + a*(-1 + b*x)*y''[x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved