## Introduction to ODEs in Higher Orders

A slecture by Yijia Wen

### 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.

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Recent Math PhD now doing a post-doctorate at UC Riverside.

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