Internal problem ID [9888]
Internal file name [OUTPUT/8835_Monday_June_06_2022_05_35_44_AM_48217882/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1566.
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.
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y=0} \] 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 (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{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 {2 \mu ^{2}+2 \nu ^{2}-4 x^{2}-7}{x^{2}}, P_{4}\left (x \right )=-\frac {2 \mu ^{2}+2 \nu ^{2}-16 x^{2}-1}{x^{3}}, P_{5}\left (x \right )=\frac {\mu ^{4}-2 \mu ^{2} \nu ^{2}+\nu ^{4}+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}}}=-2 \mu ^{2}-2 \nu ^{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}}}=-2 \mu ^{2}-2 \nu ^{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}}}=\mu ^{4}-2 \mu ^{2} \nu ^{2}+\nu ^{4} \\ {} & \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 (2 \mu ^{2}+2 \nu ^{2}-4 x^{2}-7\right ) \left (\frac {d}{d x}y^{\prime }\right )-x \left (2 \mu ^{2}+2 \nu ^{2}-16 x^{2}-1\right ) y^{\prime }+\left (\mu ^{4}-2 \mu ^{2} \nu ^{2}+\nu ^{4}+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 (\mu -\nu +r \right ) \left (-\mu -\nu +r \right ) \left (\mu +\nu +r \right ) \left (-\mu +\nu +r \right ) x^{r}+a_{1} \left (1+\mu -\nu +r \right ) \left (1-\mu -\nu +r \right ) \left (1+\mu +\nu +r \right ) \left (1-\mu +\nu +r \right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k +\mu -\nu +r \right ) \left (k -\mu -\nu +r \right ) \left (k +\mu +\nu +r \right ) \left (k -\mu +\nu +r \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 (\mu -\nu +r \right ) \left (-\mu -\nu +r \right ) \left (\mu +\nu +r \right ) \left (-\mu +\nu +r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\mu -\nu , -\mu +\nu , \mu -\nu , \mu +\nu \right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (1+\mu -\nu +r \right ) \left (1-\mu -\nu +r \right ) \left (1+\mu +\nu +r \right ) \left (1-\mu +\nu +r \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 +\mu -\nu +r \right ) \left (k -\mu -\nu +r \right ) \left (k +\mu +\nu +r \right ) \left (k -\mu +\nu +r \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+\mu -\nu +r \right ) \left (k +2-\mu -\nu +r \right ) \left (k +2+\mu +\nu +r \right ) \left (k +2-\mu +\nu +r \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+\mu -\nu +r \right ) \left (k +2-\mu -\nu +r \right ) \left (k +2+\mu +\nu +r \right ) \left (k +2-\mu +\nu +r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\mu -\nu \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k -\mu -\nu +2\right ) \left (-\nu +1-\mu +k \right )}{\left (k +2-2 \nu \right ) \left (k +2-2 \mu -2 \nu \right ) \left (k +2\right ) \left (k +2-2 \mu \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\mu -\nu \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\mu -\nu }, a_{k +2}=-\frac {4 a_{k} \left (k -\mu -\nu +2\right ) \left (-\nu +1-\mu +k \right )}{\left (k +2-2 \nu \right ) \left (k +2-2 \mu -2 \nu \right ) \left (k +2\right ) \left (k +2-2 \mu \right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\mu +\nu \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k -\mu +\nu +2\right ) \left (\nu +1-\mu +k \right )}{\left (k +2\right ) \left (k +2-2 \mu \right ) \left (k +2+2 \nu \right ) \left (k +2-2 \mu +2 \nu \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\mu +\nu \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\mu +\nu }, a_{k +2}=-\frac {4 a_{k} \left (k -\mu +\nu +2\right ) \left (\nu +1-\mu +k \right )}{\left (k +2\right ) \left (k +2-2 \mu \right ) \left (k +2+2 \nu \right ) \left (k +2-2 \mu +2 \nu \right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\mu -\nu \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k +\mu -\nu +2\right ) \left (-\nu +1+\mu +k \right )}{\left (k +2+2 \mu -2 \nu \right ) \left (k +2-2 \nu \right ) \left (k +2+2 \mu \right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\mu -\nu \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\mu -\nu }, a_{k +2}=-\frac {4 a_{k} \left (k +\mu -\nu +2\right ) \left (-\nu +1+\mu +k \right )}{\left (k +2+2 \mu -2 \nu \right ) \left (k +2-2 \nu \right ) \left (k +2+2 \mu \right ) \left (k +2\right )}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\mu +\nu \\ {} & {} & a_{k +2}=-\frac {4 a_{k} \left (k +\mu +\nu +2\right ) \left (\nu +1+\mu +k \right )}{\left (k +2+2 \mu \right ) \left (k +2\right ) \left (k +2+2 \mu +2 \nu \right ) \left (k +2+2 \nu \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\mu +\nu \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\mu +\nu }, a_{k +2}=-\frac {4 a_{k} \left (k +\mu +\nu +2\right ) \left (\nu +1+\mu +k \right )}{\left (k +2+2 \mu \right ) \left (k +2\right ) \left (k +2+2 \mu +2 \nu \right ) \left (k +2+2 \nu \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 -\mu -\nu }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k -\mu +\nu }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{k +\mu -\nu }\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k +\mu +\nu }\right ), a_{k +2}=-\frac {4 a_{k} \left (k -\mu -\nu +2\right ) \left (-\nu +1-\mu +k \right )}{\left (k +2-2 \nu \right ) \left (k +2-2 \mu -2 \nu \right ) \left (k +2\right ) \left (k +2-2 \mu \right )}, a_{1}=0, b_{k +2}=-\frac {4 b_{k} \left (k -\mu +\nu +2\right ) \left (\nu +1-\mu +k \right )}{\left (k +2\right ) \left (k +2-2 \mu \right ) \left (k +2+2 \nu \right ) \left (k +2-2 \mu +2 \nu \right )}, b_{1}=0, c_{k +2}=-\frac {4 c_{k} \left (k +\mu -\nu +2\right ) \left (-\nu +1+\mu +k \right )}{\left (k +2+2 \mu -2 \nu \right ) \left (k +2-2 \nu \right ) \left (k +2+2 \mu \right ) \left (k +2\right )}, c_{1}=0, d_{k +2}=-\frac {4 d_{k} \left (k +\mu +\nu +2\right ) \left (\nu +1+\mu +k \right )}{\left (k +2+2 \mu \right ) \left (k +2\right ) \left (k +2+2 \mu +2 \nu \right ) \left (k +2+2 \nu \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)*(4*nu^2-4*x^2-1)*y(x 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: 0.016 (sec). Leaf size: 35
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+(-2*mu^2-2*nu^2+7)*x^2)*diff(diff(y(x),x),x)+(16*x^3+(-2*mu^2-2*nu^2+1)*x)*diff(y(x),x)+(8*x^2+(mu^2-nu^2)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\mu , x\right ) c_{2} +c_{1} \operatorname {BesselJ}\left (\mu , x\right )\right ) \operatorname {BesselJ}\left (\nu , x\right )+\operatorname {BesselY}\left (\nu , x\right ) \left (\operatorname {BesselY}\left (\mu , x\right ) c_{4} +c_{3} \operatorname {BesselJ}\left (\mu , x\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.506 (sec). Leaf size: 237
DSolve[((\[Mu]^2 - \[Nu]^2)^2 + 8*x^2)*y[x] + ((1 - 2*\[Mu]^2 - 2*\[Nu]^2)*x + 16*x^3)*y'[x] + ((7 - 2*\[Mu]^2 - 2*\[Nu]^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]
\[ y(x)\to x^{-\mu -\nu } \left (c_1 \, _2F_3\left (-\frac {\mu }{2}-\frac {\nu }{2}+\frac {1}{2},-\frac {\mu }{2}-\frac {\nu }{2}+1;1-\mu ,1-\nu ,-\mu -\nu +1;-x^2\right )+c_2 x^{2 \mu } \, _2F_3\left (\frac {\mu }{2}-\frac {\nu }{2}+\frac {1}{2},\frac {\mu }{2}-\frac {\nu }{2}+1;\mu +1,1-\nu ,\mu -\nu +1;-x^2\right )+x^{2 \nu } \left (c_3 \, _2F_3\left (-\frac {\mu }{2}+\frac {\nu }{2}+\frac {1}{2},-\frac {\mu }{2}+\frac {\nu }{2}+1;1-\mu ,\nu +1,-\mu +\nu +1;-x^2\right )+c_4 x^{2 \mu } \, _2F_3\left (\frac {\mu }{2}+\frac {\nu }{2}+\frac {1}{2},\frac {\mu }{2}+\frac {\nu }{2}+1;\mu +1,\nu +1,\mu +\nu +1;-x^2\right )\right )\right ) \]