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ODE, OLDER VERSION - Coggle Diagram

- - - - Characteristic equation
        
        THEOREM: Distinct real roots
        
        If r1 and r2 are real and distinct, then this is a general solution
        
        THEOREM: Repeated real roots
        
        If r1=r2, then this is a general solution
        
        Complex Roots
        
        Assume y=e^(rx) is a solution. Therefore we have: and factor out the e^(rx) #
  - - - THEOREM: Principle of Superposition for Homogeneous Linear Equations
        
        (Informally) States that the linear combination y=c1y1+c2y2 is also a solution if y1 and y2 are solutions
      - THEOREM: General Solutions of Homogeneous Equations
        
        Assume y1 and y2 are linearly independent. If Y is a solution on the interval, then there exists c1, c2 such that Y(x)=c1y1+c2y2
    - - (Informally) States that there's a unique solution on the interval that satisfies the initial conditions y(a)=b0 and y'(a)=b1
    - - (Informally) States that
        IF
        we assume there to be a particular solution yp
        and y1 & y2 are linearly independent solutions to the associated homogeneous equation
        i.e. we have the complementary (solution) function yc=c1y1+c2y2
        THEN
        the general solution is y=yc+yp
    - - What might f(x) be?
        
        a linear combination of finite products of functions of the following types:
        
        a polynomial in x
        
        an exponential e^(rx)
        
        cos(kx) or sin(kx)
      - What are the two broad situations that may happen when attempting this method?
        
        Rule 1 or if none of f or its derivatives satisfy Ly=0
        Suppose that no term appearing either in f(x) or in any of its derivatives satisfies the associated homogeneous equation Ly = O. Then take as a trial solution for Yp a linear combination of all linearly independent such terms and their derivatives. Then determine the coefficients by substitution of this trial solution into the nonhomogeneous equation Ly = f(x).
        NOTE: usually want to use Yc to check the assumption here about f(x)
        
        Rule 2 or if one of f or its derivatives satisfy Ly=0
        ....fill in later....