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DifferentialEquations - Coggle Diagram

- - - - Practical use
        
        How fast does a disease spread
        
        Hos fast does a population change?
      - An equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one or more independent variables, is said to be a differential equation (DE).
      - Classification by type
        
        ODE - Ordinary Differential
        Equation
        
        if a differential equation contains only ordinary derivatives of one or more unknown functions with respect to a single independent variable.
        
        PDE - Partial Diferential
        Equation
        
        An equation involving partial
        derivatives of one or more unknown functions of two oe more independent variables.
        
        Partial derivatives are denoted by a subscript notation.
        
        becomes
        
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      - Notation
        
        Leibniz notation
        
        dy/dx, d²y/dx², d³y/dx³...
        
        It clearly displays both the dependent and
        independent variables.
        
        Prime notation
        
        y', y'', y''', y⁽⁴⁾...
        
        easier
        
        Newton's Dot Notation
        
        (¨flyspeck¨)
        
        used to denote derivatives with respect to time t.
        
        becomes
        
        1 more item...
      - Classification by order
        
        It is the order of the highest derivative in the equation
        
        first-order
        
        written in differential form
        
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        normal form
        
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        second-order
        
        normal form
        
        1 more item...
        
        nth-order ordinary
        
        F(x, y , y ', y'' ...., y⁽n⁾) = 0
        
        1 more item...
      - Classification by Linearity
        
        An nth-order ordinary differential equation is to be linear if F is linear in y, y',...y⁽n⁾
        
        eq (6) in page 4
        
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        Two important special cases of (6) are linear first-order (n = 1) and linear second-
        order (n = 2) DEs:
        
        eq (7) in page 4
        
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        A non-linear ordinary differrential equation is simply one that is not linear. Nonlinear functions of the dependent variable or its derivatives, such as sin y or e^y', cannot appear in a linear equation.
- - - - The derivative dy/dx of a function y ␾(x) is itself another function ␾(x)