Use the matrix exponential to solve the system X'=AX where...

Question

Use the matrix exponential to solve the system X'=AX where...

4 -3

A= ( )

3 4

Answer 1

Let x₁(0) = initial condition for X₁

x₂(0) = initial condition for X₂

The equations from the matrix are

X₁' = 4X₁ - 3X₂

X₂' = 3X₁ + 4X₂

When you solve the equations simultaneously, you obtain the following for X₁ & X₂

X₁ = (e^4xcos3x)x₁(0) - (e^4xsin3x)x₂(0)

X₂ = (e^4xsin3x)x₁(0) + (e^4xcos3x)x₂(0)