Internal problem ID [9887]
Internal file name [OUTPUT/8834_Monday_June_06_2022_05_35_37_AM_27032222/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1565.
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.
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+6 x^{3} \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\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 } \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {6}{x}, P_{3}\left (x \right )=-\frac {\rho ^{2}+\sigma ^{2}-4 x^{2}-7}{x^{2}}, P_{4}\left (x \right )=-\frac {\rho ^{2}+\sigma ^{2}-16 x^{2}-1}{x^{3}}, P_{5}\left (x \right )=\frac {\rho ^{2} \sigma ^{2}+8 x^{2}}{x^{4}}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=6 \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-\rho ^{2}-\sigma ^{2}+7 \\ {} & \circ & x^{3}\cdot P_{4}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{3}\cdot P_{4}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-\rho ^{2}-\sigma ^{2}+1 \\ {} & \circ & x^{4}\cdot P_{5}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{4}\cdot P_{5}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\rho ^{2} \sigma ^{2} \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+6 x^{3} \left (\frac {d}{d x}y^{\prime \prime }\right )-x^{2} \left (\rho ^{2}+\sigma ^{2}-4 x^{2}-7\right ) \left (\frac {d}{d x}y^{\prime }\right )-x \left (\rho ^{2}+\sigma ^{2}-16 x^{2}-1\right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \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 =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \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 =2..4 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -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 =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{3}\cdot \left (\frac {d}{d x}y^{\prime \prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{3}\cdot \left (\frac {d}{d x}y^{\prime \prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) \left (k +r -2\right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{4}\cdot \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{4}\cdot \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) \left (k +r -2\right ) \left (k +r -3\right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (r +\sigma \right ) \left (r -\sigma \right ) \left (r +\rho \right ) \left (r -\rho \right ) x^{r}+a_{1} \left (1+r +\sigma \right ) \left (1+r -\sigma \right ) \left (1+r +\rho \right ) \left (1+r -\rho \right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k +r +\sigma \right ) \left (k +r -\sigma \right ) \left (k +r +\rho \right ) \left (k +r -\rho \right )+4 a_{k -2} \left (k +r \right ) \left (k +r -1\right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (r +\sigma \right ) \left (r -\sigma \right ) \left (r +\rho \right ) \left (r -\rho \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\rho , \sigma , -\rho , -\sigma \right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (1+r +\sigma \right ) \left (1+r -\sigma \right ) \left (1+r +\rho \right ) \left (1+r -\rho \right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k} \left (k +r +\sigma \right ) \left (k +r -\sigma \right ) \left (k +r +\rho \right ) \left (k +r -\rho \right )+4 a_{k -2} \left (k +r \right ) \left (k +r -1\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & a_{k +2} \left (k +2+r +\sigma \right ) \left (k +2+r -\sigma \right ) \left (k +2+r +\rho \right ) \left (k +2+r -\rho \right )+4 a_{k} \left (k +r +2\right ) \left (k +r +1\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k +r +2\right ) \left (k +r +1\right )}{\left (k +2+r +\sigma \right ) \left (k +2+r -\sigma \right ) \left (k +2+r +\rho \right ) \left (k +2+r -\rho \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\rho \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k +\rho +2\right ) \left (k +\rho +1\right )}{\left (k +2+\rho +\sigma \right ) \left (k +2+\rho -\sigma \right ) \left (k +2+2 \rho \right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\rho \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\rho }, a_{k +2}=-\frac {4 a_{k} \left (k +\rho +2\right ) \left (k +\rho +1\right )}{\left (k +2+\rho +\sigma \right ) \left (k +2+\rho -\sigma \right ) \left (k +2+2 \rho \right ) \left (k +2\right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\sigma \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k +\sigma +2\right ) \left (k +\sigma +1\right )}{\left (k +2+2 \sigma \right ) \left (k +2\right ) \left (k +2+\rho +\sigma \right ) \left (k +2+\sigma -\rho \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\sigma \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\sigma }, a_{k +2}=-\frac {4 a_{k} \left (k +\sigma +2\right ) \left (k +\sigma +1\right )}{\left (k +2+2 \sigma \right ) \left (k +2\right ) \left (k +2+\rho +\sigma \right ) \left (k +2+\sigma -\rho \right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\rho \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k -\rho +2\right ) \left (k -\rho +1\right )}{\left (k +2+\sigma -\rho \right ) \left (k +2-\rho -\sigma \right ) \left (k +2\right ) \left (k +2-2 \rho \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\rho \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\rho }, a_{k +2}=-\frac {4 a_{k} \left (k -\rho +2\right ) \left (k -\rho +1\right )}{\left (k +2+\sigma -\rho \right ) \left (k +2-\rho -\sigma \right ) \left (k +2\right ) \left (k +2-2 \rho \right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\sigma \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k -\sigma +2\right ) \left (k -\sigma +1\right )}{\left (k +2\right ) \left (k +2-2 \sigma \right ) \left (k +2+\rho -\sigma \right ) \left (k +2-\rho -\sigma \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\sigma \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\sigma }, a_{k +2}=-\frac {4 a_{k} \left (k -\sigma +2\right ) \left (k -\sigma +1\right )}{\left (k +2\right ) \left (k +2-2 \sigma \right ) \left (k +2+\rho -\sigma \right ) \left (k +2-\rho -\sigma \right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\rho }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\sigma }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{k -\rho }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k -\sigma }\right ), a_{k +2}=-\frac {4 a_{k} \left (k +\rho +2\right ) \left (k +\rho +1\right )}{\left (k +2+\rho +\sigma \right ) \left (k +2+\rho -\sigma \right ) \left (k +2+2 \rho \right ) \left (k +2\right )}, a_{1}=0, b_{k +2}=-\frac {4 b_{k} \left (k +\sigma +2\right ) \left (k +\sigma +1\right )}{\left (k +2+2 \sigma \right ) \left (k +2\right ) \left (k +2+\rho +\sigma \right ) \left (k +2+\sigma -\rho \right )}, b_{1}=0, c_{k +2}=-\frac {4 c_{k} \left (k -\rho +2\right ) \left (k -\rho +1\right )}{\left (k +2+\sigma -\rho \right ) \left (k +2-\rho -\sigma \right ) \left (k +2\right ) \left (k +2-2 \rho \right )}, c_{1}=0, d_{k +2}=-\frac {4 d_{k} \left (k -\sigma +2\right ) \left (k -\sigma +1\right )}{\left (k +2\right ) \left (k +2-2 \sigma \right ) \left (k +2+\rho -\sigma \right ) \left (k +2-\rho -\sigma \right )}, d_{1}=0\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 Multiplying solutions by`, exp(Int(1/x, x))`Equation is the symmetric product of`, diff(diff(y(x), x), x)-(1/4)*(rho^2+2*rho*sigma+s 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 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`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 71
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+(4*x^4+(-rho^2-sigma^2+7)*x^2)*diff(diff(y(x),x),x)+(16*x^3+(-rho^2-sigma^2+1)*x)*diff(y(x),x)+(rho^2*sigma^2+8*x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right ) c_{2} +c_{1} \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ) \operatorname {BesselJ}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right )+\operatorname {BesselY}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \left (\operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right ) c_{4} +c_{3} \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.417 (sec). Leaf size: 242
DSolve[(rho^2*sigma^2 + 8*x^2)*y[x] + ((1 - rho^2 - sigma^2)*x + 16*x^3)*y'[x] + ((7 - rho^2 - sigma^2)*x^2 + 4*x^4)*y''[x] + 6*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 x^{-\rho } \, _2F_3\left (\frac {1}{2}-\frac {\rho }{2},1-\frac {\rho }{2};1-\rho ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,-\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right )+c_3 x^{-\sigma } \, _2F_3\left (\frac {1}{2}-\frac {\sigma }{2},1-\frac {\sigma }{2};1-\sigma ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}-\frac {\sigma }{2}+1;-x^2\right )+c_4 x^{\sigma } \, _2F_3\left (\frac {\sigma }{2}+\frac {1}{2},\frac {\sigma }{2}+1;-\frac {\rho }{2}+\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1,\sigma +1;-x^2\right )+c_2 x^{\rho } \, _2F_3\left (\frac {\rho }{2}+\frac {1}{2},\frac {\rho }{2}+1;\rho +1,\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right ) \]