(Created page with "== <div style="text-align:center"> Introduction to ODEs in Higher Orders </div> == <div style="text-align:center"> A slecture by Yijia Wen </div> === <small> 5.0 Abstract <sm...") |
|||

Line 6: | Line 6: | ||

− | A | + | A direct idea to deal with ODEs in higher orders is to convert them into a linear system of ODEs, which is what we are focusing at in this short tutorial. Other solutions like Laplace transforms, variation of constants and Cauchy-Euler equations will come up later. </font> |

## Revision as of 23:46, 19 November 2017

## Introduction to ODEs in Higher Orders

### 5.0 Abstract

In last tutorial we looked at three basic methods to solve differential equations in the first order. In a linear equation, we can switch the variable $ x $ to a higher order, like $ x^2 $, $ x^3 $, ..., $ x^n $ to obtain higher-ordered equations. Similarly, the differential term $ \frac{dy}{dx} $ can also be switched as $ \frac{d^2y}{dx^2} $, $ \frac{d^3y}{dx^3} $, ..., $ \frac{d^ny}{dx^n} $. This gives us the basic idea of differential equations in higher orders, the most general form for which is like $ f_n(t)\frac{d^ny}{dt^n}+f_{n-1}\frac{d^{n-1}y}{dt^{n-1}}+...+f_1(t)\frac{dy}{dt}+f_0(t)y=g(t) $, where $ n $ is the order.

A direct idea to deal with ODEs in higher orders is to convert them into a linear system of ODEs, which is what we are focusing at in this short tutorial. Other solutions like Laplace transforms, variation of constants and Cauchy-Euler equations will come up later.

** 5.5 References **

Institute of Natural and Mathematical Science, Massey University. (2017). *160.204 Differential Equations I: Course materials.* Auckland, New Zealand.

Robinson, J. C. (2003). *An introduction to ordinary differential equations.* New York, NY., USA: Cambridge University Press.

Schaft, A. J. (1986). On Realisation of Nonlinear Systems Described by Higher-Order Differential Equations. *Mathematical Systems Theory, 19* (1), p.239-275. DOI: https://doi.org/10.1007/BF01704916.