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<div style="text-align:center"> A slecture by Yijia Wen </div> | <div style="text-align:center"> A slecture by Yijia Wen </div> | ||
| − | === <small> | + | === <small> 5.0 Abstract <small> === |
<font size="3px"> 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 <math>x</math> to a higher order, like <math>x^2</math>, <math>x^3</math>, ..., <math>x^n</math> to obtain higher-ordered equations. Similarly, the differential term <math>\frac{dy}{dx}</math> can also be switched as <math>\frac{d^2y}{dx^2}</math>, <math>\frac{d^3y}{dx^3}</math>, ..., <math>\frac{d^ny}{dx^n}</math>. This gives us the basic idea of differential equations in higher orders, the most general form for which is like <math>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)</math>, where <math>n</math> is the order. | <font size="3px"> 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 <math>x</math> to a higher order, like <math>x^2</math>, <math>x^3</math>, ..., <math>x^n</math> to obtain higher-ordered equations. Similarly, the differential term <math>\frac{dy}{dx}</math> can also be switched as <math>\frac{d^2y}{dx^2}</math>, <math>\frac{d^3y}{dx^3}</math>, ..., <math>\frac{d^ny}{dx^n}</math>. This gives us the basic idea of differential equations in higher orders, the most general form for which is like <math>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)</math>, where <math>n</math> is the order. | ||
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| + | A general idea to deal with differential equations 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> | ||
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| + | ''' <big><big><big> 5.5 References </big></big></big> ''' | ||
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| + | <font size="3px"> Institute of Natural and Mathematical Science, Massey University. (2017). ''160.204 Differential Equations I: Course materials.'' Auckland, New Zealand. | ||
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| + | Robinson, J. C. (2003). ''An introduction to ordinary differential equations.'' New York, NY., USA: Cambridge University Press. | ||
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| + | 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. | ||
| + | </font> | ||
Latest revision as of 02:01, 17 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 general idea to deal with differential equations 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.
