Nth Order ODEs

Nth Order ODEs

1. Constant Coefficients?

   • Standard Forms
     • an y (n) + an−1 y (n−1) + … + a2 y″ + a1 y′ + a0 y = 0, an ≠ 0.
       • Homogeneous
         • -Find n independent solutions
           • Form a General Solutions
             • Use IC's to find constants
     • an y (n) + an−1 y (n−1) + … + a2 y″ + a1 y′ + a0 y = F(x), an ≠ 0.
       • Non Homogeneous
         • Method of undetermined coefficients
           • Use if:
             • f(x) is a polynomial, exponential, sin, cos, or any linear product or sum of them
   • Find Characteristic Equation
     • Use Rule to find a general solution
       • Use IC's

2. Variation of Parameters?

   • (1) y"+ P(x)y' +Q(x) =f(x)
     • (2) (1) y"+ P(x)y' = 0
       • yp= u1(x)y1(x) + u2(x)y2(x)
         • Solve for u1' and u2'
           • Integrate to get u1 and u2

3. Roots

   • Real Distinct, Real Equal, Complex