Internal problem ID [9892]
Internal file name [OUTPUT/8839_Monday_June_06_2022_05_36_18_AM_85937127/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1570.
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.
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+\left (6-4 a -4 c \right ) x^{3} \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
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 <- MeijerG function solution successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 49
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(6-4*a-4*c)*x^3*diff(diff(diff(y(x),x),x),x)+(-2*nu^2*c^2+2*a^2+4*(a+c-1)^2+4*(a-1)*(c-1)-1)*x^2*diff(diff(y(x),x),x)+(2*nu^2*c^2-2*a^2-(2*a-1)*(2*c-1))*(2*a+2*c-1)*x*diff(y(x),x)+((-c^2*nu^2+a^2)*(-c^2*nu^2+a^2+4*a*c+4*c^2)-b^4*c^4*x^(4*c))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (\nu , b \,x^{c}\right ) c_{1} +\operatorname {BesselY}\left (\nu , b \,x^{c}\right ) c_{2} +\operatorname {BesselY}\left (\nu , i b \,x^{c}\right ) c_{4} +\operatorname {BesselJ}\left (\nu , i b \,x^{c}\right ) c_{3} \right ) x^{a} \]
✓ Solution by Mathematica
Time used: 0.102 (sec). Leaf size: 213
DSolve[((a^2 - c^2*\[Nu]^2)*(a^2 + 4*a*c + 4*c^2 - c^2*\[Nu]^2) - b^4*c^4*x^(4*c))*y[x] + (-1 + 2*a + 2*c)*(-2*a^2 - (-1 + 2*a)*(-1 + 2*c) + 2*c^2*\[Nu]^2)*x*y'[x] + (-1 + 2*a^2 + 4*(-1 + a)*(-1 + c) + 4*(-1 + a + c)^2 - 2*c^2*\[Nu]^2)*x^2*y''[x] + (6 - 4*a - 4*c)*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to b^{a/c} (-1)^{\frac {a-c \nu }{4 c}} 2^{-\frac {2 a}{c}-\nu -3} \left (x^{4 c}\right )^{\frac {a}{4 c}} \left (4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )-i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselJ}\left (-\nu ,b \sqrt [4]{x^{4 c}}\right )+4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )+i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselI}\left (-\nu ,b \sqrt [4]{x^{4 c}}\right )+i^{\nu } \left ((4 c_3 \operatorname {Gamma}(\nu +1)-i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselJ}\left (\nu ,b \sqrt [4]{x^{4 c}}\right )+(4 c_3 \operatorname {Gamma}(\nu +1)+i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselI}\left (\nu ,b \sqrt [4]{x^{4 c}}\right )\right )\right ) \]