ECE 205

ECE 205

1. Ordinary Differential Equation

   • Only contain singular derivatives
     • Linear ODEs
       • First Order
         • Homogenous Case
           • dy/dt + p(t)y = 0
             • Solution: Ae^-int(p(t) dt), where A is any constant
         • Non-homogenous
           • dy/dt + p(t)y = q(t)
             • Solution: e^-int(p(t) dt) * [int(e^int(p(t) dt) * q(t) dt]
       • Nth Order
         • Homogenous
           • Has n linearly independent solutions which can be combined to make other solutions
           • Constant Coefficients
             • Solution: Try e^(lambda t), substitute derivatives for equation
               • Case 1: Distinct real roots
                 • Each distinct root is added to the solution
               • Case 2: Complex conjugate pair a+-jb
                 • Each distinct half of the pair is added to the solution
               • Case 3: Double Root
                 • Multiply term by t and add it to the end for every extra term you need
           • Non Constant Coefficients
             • Cauchy Euler - t^ny^n + c1t^(n-1)y^(n-1)...cny = 0
               • Solution: Try t^lambda, sub derivatives for equation, d = root((1-c1)^2 - 4c2)
                 • Case 1: d > 0, distinct roots
                   • t^lambdak is a LI solution
                 • Case 2: d = 0, repeated
                   • repeated M times
                     • [D1 + D2ln|t| +...+Dm(ln|t|)^m-1]t^lambdak
                 • Case 3: d < 0, complex roots
                   • Use complex roots, turn to real if needed
         • Particular
           • y(t) = hom soln+ part soln
             • if the DE is of the form (y and its derivatives) = f1(t) + f2(t), solution is homo+part1+part2
               • You can create multiple general solutions by adding any particular solution to the homogenous solution
           • Method of Undetermined Coefficients
             • Take derivatives until we don't get LI solutions
               • Solution is some combination of those
     • Nonlinear ODEs
       • Separable Equations
         • dy/dt = T(t) Y(y)
           • Solution: Move all y terms on one side, t terms on other, and integrate
             • int(1/Y(y) dy) = int(T(t) dt)
               • Solve for y.

2. Partial Differential Equations

   • Contain Partial Derivatives

3. Laplace Transforms

   • Only apply to causal functions
   • F(s) = int0->inf( f(t) e^-st dt), where s is a complex number
   • Linear, so use that property to go backwards
     • Manipulate a function in the complex domain in such a way that allows us to recover functions we know from the transform tables