Internal problem ID [9817]
Internal file name [OUTPUT/8760_Monday_June_06_2022_05_24_39_AM_48217276/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1491.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime \prime \prime }+3 y^{\prime \prime } x +\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime }-4 a^{3} x^{2 a -1} y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime \prime }\right )+3 \left (\frac {d}{d x}y^{\prime }\right ) x +\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime }-4 a^{3} x^{2 a -1} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third 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: 3; missing the dependent variable -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius <- pFq successful: received ODE is equivalent to the 1F2 ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 88
dsolve(x^2*diff(y(x),x$3)+3*x*diff(y(x),x$2)+(4*a^2*x^(2*a)+1-4*nu^2*a^2)*diff(y(x),x)=4*(a^3*x^(2*a-1))*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [\nu +1, -\nu +1\right ], -x^{2 a}\right )+c_{2} x^{-2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\nu \right ], \left [-\nu +1, 1-2 \nu \right ], -x^{2 a}\right )+c_{3} x^{2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\nu \right ], \left [\nu +1, 2 \nu +1\right ], -x^{2 a}\right ) \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 102
DSolve[(1 - 4*a^2*nu^2 + 4*a^2*x^(2*a))*y'[x] + 3*x*y''[x] + x^2*Derivative[3][y][x] == 4*a^3*x^(-1 + 2*a)*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \left (x^{2 a}\right )^{-\nu } \, _1F_2\left (-\nu -\frac {1}{2};1-2 \nu ,1-\nu ;-x^{2 a}\right )+c_3 \left (x^{2 a}\right )^{\nu } \, _1F_2\left (\nu -\frac {1}{2};\nu +1,2 \nu +1;-x^{2 a}\right )+c_1 \, _1F_2\left (-\frac {1}{2};1-\nu ,\nu +1;-x^{2 a}\right ) \]