Solving Second Order Ordinary Differential Equations

Homogenous

Consider the function . Here is a constant to be found.

By applying the function to the above equation:

The above equation is called the Auxiliary equation or Characteristic equation.

Case 1: Distinct real roots

Case 2: Equal real roots

Case 3: Complex conjugate roots

Non-homogenous

Method of undetermined coefficients

is guessed with undetermined coefficients. And subtituted in the given equation to find the coefficients. The guess depends on the nature of .

If is:

• a constant, is a constant
• ,
• ,
• or ,
• , (Only works if is not a root of auxiliary equation)
• A product of and some , guess for individually, and then multiply by (without coefficients)
• A product of polynomials and trig functions, guess for the polynomial, and multiply that by the appropriate cosine. Then add on a new guess for the polynomial with different coefficients and multiply that by the appropriate sine.
• A sum of functions, can be guessed individually and be summed up

Steps

• Solve for
• Based on the form of , make an initial guess for .
• Check if any term in the guess for is a solution to the complementary equation.
• If so, multiply the guess by . Repeat this step until there are no terms in that solve the complementary equation.
• Substitute into the differential equation and equate like terms to find values for the unknown coefficients in .
• If coefficients were unable to be found (they cancelled out or something like that), multiply the guess by and start again.