Question

if φ(t) is a solution of x¨=Ax where A is a constant matrix with red-valued entries, then the real part of φ(t)(Re(φ(t)))& the imaginary part of φ(t)(Im(φ(t))) are both solutims to x˙=Ax

Answer 1

The real and imaginary parts of the solution φ(t) to x¨=Ax are both solutions to x˙=Ax.

Step-by-step explanation:

If φ(t) is a solution of x¨=Ax where A is a constant matrix with real-valued entries, then the real part of φ(t) (Re(φ(t))) and the imaginary part of φ(t) (Im(φ(t))) are both solutions to x˙=Ax.

To prove this, we can substitute φ(t) into the differential equation x˙=Ax and calculate the first derivative of Re(φ(t)) and Im(φ(t)). We will find that they satisfy the equation and hence are solutions.

Therefore, both the real and imaginary parts of φ(t) satisfy the equation x˙=Ax.