Internal problem ID [9797]
Internal file name [OUTPUT/8740_Monday_June_06_2022_05_22_24_AM_65875645/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1471.
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.
\[ \boxed {y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime }+f \left (x \right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+f \left (x \right ) \left (\frac {d}{d x}y^{\prime }\right )+y^{\prime }+f \left (x \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
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 trying a solution in terms of MeijerG functions -> Try computing a Rational Normal Form for the given ODE... <- unable to resolve the Equivalence to a Rational Normal Form trying reduction of order using simple exponentials -> Calling odsolve with the ODE`, diff(_b(_a), _a) = (I-f(_a))*_b(_a), _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- reduction of order using simple exponentials successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve(diff(diff(diff(y(x),x),x),x)+f(x)*diff(diff(y(x),x),x)+diff(y(x),x)+f(x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{i x} \left (\int {\mathrm e}^{-2 i x} \left (c_{3} \left (\int {\mathrm e}^{\int \left (i-f \left (x \right )\right )d x}d x \right )+c_{2} \right )d x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.063 (sec). Leaf size: 84
DSolve[f[x]*y[x] + y'[x] + f[x]*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_3 e^{i x} \int _1^xe^{-2 i K[3]} \int _1^{K[3]}\exp \left (\int _1^{K[2]}(i-f(K[1]))dK[1]\right )dK[2]dK[3]+c_1 e^{i x}+\frac {1}{2} i c_2 e^{-i x} \]