91Ó°ÊÓ

Differentiating vector functions is an essential concept in solving systems of differential equations. Imagine you have a vector whose components are functions of some variable—generally time. To differentiate this vector function, we separately differentiate each component function with respect to that variable.

Consider a vector function of the form:
\[ Y(t) = \begin{bmatrix}y_1(t) \ y_2(t) \ \vdots \ y_n(t)\end{bmatrix} \].
Its derivative with respect to time t, denoted as \( Y'(t) \), is obtained by differentiating each function within the vector:\[ Y^{\text{'}}(t) = \frac{d}{dt}\begin{bmatrix}y_1(t) \ y_2(t) \ \vdots \ y_n(t)\end{bmatrix} = \begin{bmatrix} \frac{dy_1(t)}{dt} \ \frac{dy_2(t)}{dt} \ \vdots \ \frac{dy_n(t)}{dt}\end{bmatrix} \].
This operation is similar to differentiating scalar functions, but here it's applied to each component of the vector. The result is another vector function where each entry is the derivative of the corresponding component of the original vector.

Multiplication by a constant matrix is a commonplace operation in linear algebra, particularly when it comes to solving systems of differential equations. The constant matrix acts on a vector, transforming it into another vector through a linear combination of its elements.

For a constant matrix A and a vector function Y, expressed as:
\[ A = \begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n}\ a_{21} & a_{22} & \cdots & a_{2n}\ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}, \quad Y(t) = \begin{bmatrix}y_1(t) \ y_2(t) \ \vdots \ y_n(t)\end{bmatrix} \],
the product \( AY \) is calculated by taking the linear combination of the columns of A with the elements of Y(t). This results in a new vector where each component is a sum of products of constants from A and functions from Y:
\[ AY(t) = \begin{bmatrix}a_{11}y_1(t) + a_{12}y_2(t) + \cdots + a_{1n}y_n(t) \ a_{21}y_1(t) + a_{22}y_2(t) + \cdots + a_{2n}y_n(t) \ \vdots \ a_{n1}y_1(t) + a_{n2}y_2(t) + \cdots + a_{nn}y_n(t)\end{bmatrix} \].
The significance of this operation in differential equations is that it can represent the system's dynamics where each equation describes the rate of change of one state variable, and the matrix encodes how each variable interacts with the others.

Verifying solutions to differential equations is a crucial step to ensure that the purported solution indeed fits the given equation. The verification process involves substituting the candidate solution back into the original differential equation and checking if the equality holds for all relevant values.

In our textbook example, verification means proving that \( Y' = AY \). Here, we substitute the derivative of Y(t), calculated as in the earlier section, into the left side of the equation, and the product of the matrix A and vector Y(t) into the right side of the equation. We need to confirm that for each entry in the vector, the derivative matches the corresponding entry in the resulting vector from the matrix multiplication:
\[ \frac{dy_i(t)}{dt} = a_{i1}y_1(t) + a_{i2}y_2(t) + \cdots + a_{in}y_n(t) \quad (\text{for } i = 1, 2, \ldots, n) \].
If these equations are satisfied for every i, our original assertion that \( Y' = AY \) is verified. This process is indicative of a broader practice in differential equations where solutions, often derived through various means, must always be checked for consistency with the original problem.