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ODE09H26 - ODE Basics The rest is important details...

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ODE Basics The rest is important details Stephanos Venakides August 27, 2009 Motivation This outline aims at a basic initial contact with prominent concepts and calculations of the subject of ordinary differential equations (ODE). The intention is that the student will approach additional material as an elaboration into an already familiar global view. The outline also makes a connection between the ODE material taught in MTH107 with the ODE material of MTH108. What is an ODE or a system of ODE? Ordinary differential equations and systems of such equations are often considered in a form that describes the motion of a particle in a space called phase-space . In applications, phase- space is generally not the physical, geometric space. Its coordinates may quantify physical, chemical, biological, financial, social or other entities. Similarly, the “time” variable t may not stand for time at all. It may represent some other variable on which the phase-space coordinates depend. The mathematical analysis of ODE is independent of the context of particular applications. Nevertheless, the intuition of a particle moving in physical space is helpful and is pursued here. An ordinary differential equation (or the system of equations) in phase-space form ex- presses the particle velocity in terms of its position; inserting the coordinates of the position of the particle in the equation or system produces the value of the particle’s velocity. For ex- ample, in the case of a 2-dimensional phase-space, or phase-plane , with coordinates ( y 1 , y 2 ), the ODE system is dy 1 dt = V 1 ( y 1 , y 2 ) dy 2 dt = V 2 ( y 1 , y 2 ) . (1) The functions V 1 ( y 1 , y 2 ) and V 2 ( y 1 , y 2 ), which give the velocity in terms of the position, are assumed to be given. The intuition is that, as the particle changes position, its velocity is continuously updated by the ODE and the motion is, thus, specified. A more precise version of this statement is a mathematical theorem, if sufficient smoothness of the velocity field is assumed. The smoothness condition is satisfied, for example, if the expression for velocity 1
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is continuously differentiable with respect to the coordinates of phase-space. The system of ODE in an n -dimensional phase-space is often written in vector form d y dt = V ( y ) , (2) where, y = y 1 . . y n , V = V 1 . . V n . (3) Each of the components of the velocity vector depends on the position coordinates y 1 , · · · , y n but not on time. The function V = V ( y ) is a vector field , that is “plugging-in” the vector y produces another vector, V . A large part of the material of MTH107 consists of the solution of such systems in the special case in which the velocity depends on the position linearly, d y dt = A y . (4) Here, A is a constant square matrix that multiplies the position vector y . Solving this system requires the calculation of the eigenvalues and corresponding eigenvectors of the matrix. In two dimensions, the system of equations is dy 1 dt = a 11 y 1 +
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