Internal problem ID [9830]
Internal file name [OUTPUT/8773_Monday_June_06_2022_05_26_09_AM_38233570/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1504.
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.
\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 y x=0} \] Unable to solve this ODE.
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 Equation is the LCLM of -2/x*y(x)+diff(y(x),x), y(x)+diff(diff(y(x),x),x) trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful <- solving the LCLM ode successful `
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 18
dsolve((x^2+2)*diff(diff(diff(y(x),x),x),x)-2*x*diff(diff(y(x),x),x)+(x^2+2)*diff(y(x),x)-2*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{2}+c_{2} \cos \left (x \right )+c_{3} \sin \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.166 (sec). Leaf size: 41
DSolve[-2*x*y[x] + (2 + x^2)*y'[x] - 2*x*y''[x] + (2 + x^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} \left (2 c_1 x^2+2 i c_2 e^{-i x}-c_3 e^{i x}\right ) \]