Question

Let X be a fundamental matrix for the ODE x' = A and let X₀ := X(0).

Show that Y(t) := X(t)X⁻¹ satisfies the initial value problem
Y' = AY
Y(0) = I
where I denotes the identity matrix.

Answer 1

Y(t) := X(t)X⁻¹ is shown to be a solution to the initial value problem Y' = AY with Y(0) = I by using the properties of a fundamental matrix for the differential equation x' = Ax and the product rule for differentiation.

Step-by-step explanation:

The question asks us to show that Y(t) := X(t)X⁻¹ is a solution to the initial value problem Y' = AY with Y(0) = I, where I is the identity matrix. We know that X(t) is a fundamental matrix for the differential equation x' = Ax, and X₀ := X(0) is the value of X at t=0.

To prove that Y(t) satisfies the equation Y' = AY, we differentiate Y(t) with respect to t:

1. Calculate Y' by differentiating Y(t) = X(t)X⁻¹. Using the product rule for differentiation, we get Y' = X'(t)X⁻¹ + X(t)(X⁻¹)'.
2. Since X'(t) = AX(t), substitute to find Y' = AX(t)X⁻¹.
3. Recognize that X(t)X⁻¹ is equal to Y(t) by definition, leading to Y' = AY.
4. Lastly, to confirm the initial condition, evaluate Y at t=0: Y(0) = X(0)X⁻¹(0) = X₀ X⁻¹₀ = I, since X₀ is invertible and its inverse is X⁻¹₀.

Therefore, we conclude that Y(t) satisfies the differential equation and the initial condition, making Y(t) = X(t)X⁻¹ a solution to the initial value problem Y' = AY with Y(0) = I.