Second Order Constant Coefficient Differential Vector Equations

To solve a second order constant coefficient differential equation of the formwe assume a solution of the formTo solve a second order constant coefficient differential vector equation of the formwe assume a solution of the form

Suppose we have an equation

with initial conditionsandat

Assume a solution of the formso thatandSubstitute these into the differential equation to get

We can factorise withto obtain

since ifthe initial solutions will not be satisfied, and the exponential factor is not equal to 0 for any finiteso

HenceWe have to fit this solution to the initial conditions.

at(1)

at(2)

(1)+(2) givesthen from (1)