Friday, October 26, 2012

Differential Equations - Constant Coefficients

This will go over the process of solving simple differential equations with constant coefficients by using the characteristic equation. The basic idea is to replace the order of the differential with a polynomial that has the same power and coefficients as the differential. The solution of this polynomial, depending on the sign or type of number the solution is gives us the coefficients to the general solutions we already know for this type of differential equation. Lets go over the following example

We can replace this with

r3 - 3 r2 + 6 r - 18 = 0

We then solve this equation in order to find the coefficients to a general solution.

r2 (r - 3) + 6( r - 3) = 0
(r - 3) ( r2 + 6) = 0
r = 3, +- (6)1/2i

So now that we solved this equation, we can note that theres a 3. This means that the coefficient of an exponential is 3

y = c1e3x

The complex solutions result in a different way, resulting in sin and cosine solutions. This can be proven using complex analysis and substituting in using euler's equation.

Resulting in a final combined solution of this

Which is in a form we can use in order to completely solve if we have initial conditions to find the coefficients c1, c2 and c3. Lets use the example that at y(0) = 1, y'(0) = 1, y''(0) = 1. This would mean that

After taking the derivative, then the double derivative, and plugging in for x = 0. This gives us a system of equations which we can solve by taking the reduced row echelon form of it.

Giving us a solution for all the coefficients and a final answer of

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