Lectures 1, 2 and 3

Lectures 1, 2 and 3 - MTR 500 ODEs Review M.A. Jarrah...

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MTR 500 ODEs Review M.A. Jarrah
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Dynamic Systems Classification Dynamic Static Time Invariant Time Varying Nonlinear Linear Continuous-State Discrete-State Time-Driven Event-Driven Deterministic Stochastic Discrete-Time Continuous-Time
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Differential equations are classified by Number of independent variables Number of unknown functions Order of the highest derivative Order of nonlinearity Ordinary differential equations (ODE): one independent variable Partial differential equations (PDE): two or more independent variables A single differential equation A system of differential equations First order differential equations Second order differential equations Higher order differential equations Linear differential equations Nonlinear differential equations Classification of Differential Equations
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ODE PDE System of differential equations , ( ), ( ) dx dy ax xy cy xy dt dt x x t y y t α γ = - = - + = = Lotka-Volterra equations Classification , ( ) dv m mg v v v t dt = - = Newton’s second law for the falling rock 2 2 2 2 2 , ( , ) u u a u u x t x t = = d means ordinary or full derivative Wave equation means partial derivative A single equation First order equation Second order equation Linear equation 1 0 1 1 ( ) ( ) ( ) ( ), ( ) n n n n n d y d y a t a t a t y g t y y t dt dt - - + + + = = K General linear ODE of order n 2 2 , , , , , 0, ( ) n u du d u d u F t u u u t dt dt dt = = K 2 1 2 1 , , , , , , ( ) n n n n d y dy d y d y f t y y y t dt dt dt dt - - = = K or ODE of order n Nonlinear equation Newton’s inverse square gravitational law for planetary motion 2 2 2 2 3 , ( ) d r k h r r t dt r r = - + =
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Classification Definitions Any function φ defined on some interval I , which when substituted into a differential equation reduces the equation to an identity Is said to be a solution of the equation on the interval. A solution in which the dependent variable is expressed solely in terms of the independent variable is said to be an explicit solution. An explicit solution of a differential equation that is identically zero on an interval I is said to be a trivial solution . A relation G(t,y)=0 is said to be an implicit solution of ODE on an interval I provided there exists at least one function that satisfies the relation as well as the differential equation on I . Examples 4 16 t y = is an explicit solution of the equation 1/ 2 dy ty dt = on the interval ( , ) I = -∞ ∞ 2 2 4 0 t y + - = is an implicit solution of the equation dy t dt y =- on ( 2,2) I = -
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Initial value problem 0.5 450, ( ) dp p p p t dt = - = Direction field Differential equation may have infinite number of solutions. A family of all possible solutions is said to be the general solution of the equation. Solution (integral) curves General solution / 2 900 t p Ce = + C is an arbitrary constant (parameter) of the solution A particular solution of the equation is free of arbitrary parameters (we need to define a value of C ) / 2 900 200 t p e = - If p (0) = p 0 = 700, then C = - 200 and A differential equation together with an initial condition form an initial value problem 0 0.5 450, (0) dp p dt p p = - = The value of C can be found from the initial condition: p (0) = p 0
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Lectures 1, 2 and 3 - MTR 500 ODEs Review M.A. Jarrah...

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