Basic Theory of Systems of First Order Linear Equations

[Jamin2112]

Homework Statement

[I'm going to write column vectors as the row vectors transposed, since I don't have a fancy-schmancy equation-writing program]

Consider the vectors x⁽¹⁾(t)=(t 1)^T and x⁽²⁾(t)=(t² 2t)^T

(a) Compute the Wronskian of x⁽¹⁾ and x⁽²⁾.
(b) In what intervals are x⁽¹⁾ and x⁽²⁾ linearly independent?
(c) What conclusion can be drawn about the coefficients in the system of homogenous differential equations satisfied by x⁽¹⁾ and x⁽²⁾?
(d) Find this system of equations and verify the conclusions of part (c)?

Homework Equations

The Attempt at a Solution

(a) are (b) are easy. The Wronskian it t², and since vectors are only linearly independent if the determinant ≠ 0, the interval in this case is t≠0.

Parts (c) and (d) have me stuck. Can you lead me in the right direction?

 

[mighty2000]

Thank you for your question. Let's start by defining what the Wronskian is. The Wronskian of two vectors x and y is given by the determinant |x y|. In this case, the Wronskian of x(1) and x(2) is given by |(t 1)T (t2 2t)T|. Expanding this determinant, we get t2-2t, which is indeed equal to t2 and is non-zero for all values of t except t=0. Therefore, x(1) and x(2) are linearly independent for all values of t except t=0.

Now, let's look at the coefficients in the system of homogeneous differential equations satisfied by x(1) and x(2). Since x(1) and x(2) are linearly independent, the coefficients must be non-zero for all values of t except t=0. This means that the system of equations must be of the form:
x'(t)=a(t)x(t)+b(t)x(t), where a(t) and b(t) are non-zero functions.

To verify this, we can take the derivative of x(1) and x(2) and substitute them into the above equation. We get:

x'(1)(t)=(1 0)T and x'(2)(t)=(2t 2)T. Substituting these into the equation, we get:

(1 0)T=a(t)(t 1)T + b(t)(t2 2t)T
(2t 2)T=a(t)(t 1)T + b(t)(t2 2t)T

Simplifying these equations, we get:

a(t)=1 and b(t)=2

Therefore, the system of equations satisfied by x(1) and x(2) is:

x'(t)=x(t)+2x(t)

This confirms our conclusion that the coefficients in the system of equations must be non-zero for all values of t except t=0. I hope this helps. Best of luck with your studies!