Internal problem ID [9798]
Internal file name [OUTPUT/8741_Monday_June_06_2022_05_22_31_AM_15021856/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1472.
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.
\[ \boxed {y^{\prime \prime \prime }+f \left (x \right ) \left (x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y\right )=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+f \left (x \right ) \left (x^{2} \left (\frac {d}{d x}y^{\prime }\right )-2 y^{\prime } x +2 y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
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 --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, x], [0, y], [0, x^2]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 33
dsolve(diff(diff(diff(y(x),x),x),x)+f(x)*(x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*y(x))=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\int \left (c_{1} +c_{2} \left (\int {\mathrm e}^{-\left (\int \frac {x^{3} f \left (x \right )+3}{x}d x \right )}d x \right )\right )d x +c_{3} \right ) x \]
✓ Solution by Mathematica
Time used: 0.072 (sec). Leaf size: 85
DSolve[f[x]*(2*y[x] - 2*x*y'[x] + x^2*y''[x]) + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (c_3 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}f(K[1]) K[1]^2dK[1]\right )}{K[2]^2}dK[2]-x \int _1^x\frac {\exp \left (-\int _1^{K[3]}f(K[1]) K[1]^2dK[1]\right )}{K[3]^3}dK[3]\right )+c_2 x+c_1\right ) \]