Internal problem ID [9865]
Internal file name [OUTPUT/8810_Monday_June_06_2022_05_32_10_AM_11907943/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1541.
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.
\[ \boxed {y^{\prime \prime \prime \prime }+\left (a \,x^{2}+b \lambda +c \right ) y^{\prime \prime }+\left (a \,x^{2}+\beta \lambda +\gamma \right ) y=0} \] 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 }+\left (a \,x^{2}+b \lambda +c \right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (a \,x^{2}+\beta \lambda +\gamma \right ) 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 ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )}{\sum }}a_{k} x^{k +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )+m}{\sum }}a_{k -m} 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..2 \\ {} & {} & 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 ODE with series expansions}\hspace {3pt} \\ {} & {} & 24 a_{4}+2 a_{2} \left (b \lambda +c \right )+a_{0} \left (\beta \lambda +\gamma \right )+\left (120 a_{5}+6 a_{3} \left (b \lambda +c \right )+a_{1} \left (\beta \lambda +\gamma \right )\right ) x +\left (\moverset {\infty }{\munderset {k =2}{\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 ) \left (b \lambda +c \right )+a_{k} \left (a \,k^{2}-a k +\beta \lambda +\gamma \right )+a_{k -2} a \right ) x^{k}\right )=0 \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} x \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [24 a_{4}+2 a_{2} \left (b \lambda +c \right )+a_{0} \left (\beta \lambda +\gamma \right )=0, 120 a_{5}+6 a_{3} \left (b \lambda +c \right )+a_{1} \left (\beta \lambda +\gamma \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{4}=-\frac {1}{12} a_{2} b \lambda -\frac {1}{24} a_{0} \beta \lambda -\frac {1}{12} a_{2} c -\frac {1}{24} a_{0} \gamma , a_{5}=-\frac {1}{20} a_{3} b \lambda -\frac {1}{120} a_{1} \beta \lambda -\frac {1}{20} a_{3} c -\frac {1}{120} a_{1} \gamma \right \} \\ \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 ) \left (b \lambda +c \right )+a_{k} \left (\left (k^{2}-k \right ) a +\beta \lambda +\gamma \right )+a_{k -2} a =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & a_{k +6} \left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (k +3\right )+a_{k +4} \left (k +4\right ) \left (k +3\right ) \left (b \lambda +c \right )+a_{k +2} \left (\left (\left (k +2\right )^{2}-k -2\right ) a +\beta \lambda +\gamma \right )+a_{k} a =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 +6}=-\frac {b \,k^{2} \lambda a_{k +4}+a \,k^{2} a_{k +2}+7 b k \lambda a_{k +4}+c \,k^{2} a_{k +4}+3 a k a_{k +2}+12 b \lambda a_{k +4}+\beta \lambda a_{k +2}+7 c k a_{k +4}+a_{k} a +2 a a_{k +2}+12 c a_{k +4}+\gamma a_{k +2}}{\left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (k +3\right )}, a_{4}=-\frac {1}{12} a_{2} b \lambda -\frac {1}{24} a_{0} \beta \lambda -\frac {1}{12} a_{2} c -\frac {1}{24} a_{0} \gamma , a_{5}=-\frac {1}{20} a_{3} b \lambda -\frac {1}{120} a_{1} \beta \lambda -\frac {1}{20} a_{3} c -\frac {1}{120} a_{1} \gamma \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*x^2+b*lambda+c)*diff(diff(y(x),x),x)+(a*x^2+beta*lambda+gamma)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(\[Gamma] + \[Beta]*\[Lambda] + a*x^2)*y[x] + (c + b*\[Lambda] + a*x^2)*y''[x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved