Question

In each case, show that the set of vector functions \(\phi_{1}\) and \(\phi_{2}\) defined for all \(t\) as indicated, is linearly independent on any interval \(a \leq t \leq b\). (a) \(\boldsymbol{\phi}_{1}(t)=\left(\begin{array}{c}e^{t} \\ 2 e^{t}\end{array}\right) \quad\) and \(\quad \boldsymbol{\phi}_{2}(t)=\left(\begin{array}{c}e^{s t} \\ 4 e^{3 t}\end{array}\right)\). (b) \(\phi_{1}(t)=\left(\begin{array}{c}2 e^{2 t} \\ -e^{2 t}\end{array}\right)\) and \(\phi_{2}(t)=\left(\begin{array}{c}e^{-t} \\ 3 e^{-t}\end{array}\right)\)

Short answer

Using the Wronskian method, we find: (a) The Wronskian for \(\phi_1(t)\) and \(\phi_2(t)\) is \(W = 4e^{4t} - 2e^{(s+1)t}\), which is always positive and not equal to zero on the interval \(a \leq t \leq b\). Therefore, \(\phi_1(t)\) and \(\phi_2(t)\) are linearly independent on any interval \(a \leq t \leq b\). (b) The Wronskian for \(\phi_1(t)\) and \(\phi_2(t)\) is \(W = 6e^{t} - e^{t}\), which is always positive and not equal to zero on the interval \(a \leq t \leq b\). Therefore, \(\phi_1(t)\) and \(\phi_2(t)\) are linearly independent on any interval \(a \leq t \leq b\).

Step-by-step solution

Compute the Wronskian matrix for \(\phi_1 (t)\) and \(\phi_2 (t)\)

First, write down the matrix with the given vector functions and their derivatives with respect to t: \[ \begin{bmatrix} e^t & e^{st} \\ 2e^t & 4e^{3t} \end{bmatrix} \]

Compute the determinant of the Wronskian matrix

Calculate the determinant by multiplying the diagonal entries and subtracting the product of the off-diagonal entries: \[ W = e^{t} \cdot 4e^{3t} - e^{st} \cdot 2e^t = 4e^{4t} - 2e^{(s+1)t} \]

Show that the Wronskian is not equal to zero on the interval \(a \leq t \leq b\)

Since \(e^t\) and \(e^{4t}\) are always positive for any t, the Wronskian will always be positive and not equal to zero on the interval \(a \leq t \leq b\). Therefore, \(\phi_1(t)\) and \(\phi_2(t)\) are linearly independent on any interval \(a \leq t \leq b\). (b)

Compute the Wronskian matrix for \(\phi_1 (t)\) and \(\phi_2 (t)\)

First, write down the matrix with the given vector functions and their derivatives with respect to t: \[ \begin{bmatrix} 2e^{2t} & e^{-t} \\ -e^{2t} & 3e^{-t} \end{bmatrix} \]

Compute the determinant of the Wronskian matrix

Calculate the determinant by multiplying the diagonal entries and subtracting the product of the off-diagonal entries: \[ W = (2e^{2t}) \cdot (3e^{-t}) - (e^{-t}) \cdot (-e^{2t}) = 6e^{t} - e^{t} \]

Show that the Wronskian is not equal to zero on the interval \(a \leq t \leq b\)

It can be seen that the Wronskian is always positive and not equal to zero on the interval \(a \leq t \leq b\). Therefore, \(\phi_1(t)\) and \(\phi_2(t)\) are linearly independent on any interval \(a \leq t \leq b\).

Key concepts

Wronskian Matrix

When determining the linear independence of vector functions, particularly in the context of solutions to ordinary differential equations (ODEs), the Wronskian matrix is a vital tool. This matrix is constructed using the vector functions in question and their derivatives.

For two vector functions \( \phi_1(t) \) and \( \phi_2(t) \), the Wronskian matrix is composed as follows: \[ \begin{bmatrix} \phi_1(t) & \phi_2(t) \ \frac{d}{dt}\phi_1(t) & \frac{d}{dt}\phi_2(t) \end{bmatrix} \]

Significance of the Wronskian

If the determinant of this matrix, known as the Wronskian determinant, is non-zero for all points on an interval, it implies that the vector functions are linearly independent on that interval. This is because linear dependence would mean that one vector function can be expressed as a scalar multiple of the other, leading to a determinant of zero. Consequently, the Wronskian matrix serves as a check for linear independence.

Determinant Computation

The determinant of a matrix provides valuable information about the matrix and can be particularly insightful when derived from a Wronskian matrix.

In general, for a 2x2 matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant is calculated by the formula \[ \text{det} = ad - bc \].

Application in Wronskian

When applied to the Wronskian matrix, this calculation of the determinant is used to establish the linear independence of the vector functions. If the resulting determinant, or Wronskian, is not zero on an interval, the functions do not exhibit any form of linear dependency over that range. This step is consistently used across various problems to validate the linear independence of vector functions in question.

Interval of Independence

The interval of independence refers to the range of values where two or more functions are linearly independent. For vector functions subject to an ordinary differential equation, the interval of independence is particularly pertinent since solutions to ODEs can have different behaviors over different intervals.

Finding the Interval

To determine the interval of independence, we examine where the Wronskian is nonzero. In the case of the exercise provided, the Wronskian never equals zero, signifying that the functions are linearly independent no matter which interval \( a \leq t \leq b \) we choose. This is an essential concept because it indicates that the solutions to the differential equation form a fundamental set of solutions on the interval, enabling the construction of general solutions to the differential equation.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) characterize relationships involving rates of change of a variable with respect to another variable. Solving these equations often leads to the identification of functions or sets of functions that describe dynamic systems or phenomena.

Solutions and Linear Independence

In the context of ODEs, the concept of linear independence is pivotal. The general solution to a linear differential equation is commonly expressed as a linear combination of linearly independent solutions. These solutions form a basis for the solution space of the differential equation.

• Linear dependence means a set of solutions is overlapping and redundant.
• Linear independence ensures a set of solutions is complete and sufficient to describe all possible behaviors of the system governed by the ODE.

By utilizing tools like the Wronskian matrix and understanding the interval of independence, students can effectively determine the nature of solutions to ODEs and apply these findings to model complex systems.