Question

The Wronskian of two functions is \(W(t)=t^{2}-4 .\) Are the functions linearly independent or linearly dependent? Why?

Short answer

Answer: The functions are linearly independent.

Step-by-step solution

Check if the Wronskian is non-zero for any value of t.

We are given the Wronskian W(t) = t^2 - 4. To check if this is non-zero for any value of t, we can try to find the roots (if any) of the equation t^2 - 4 = 0.

Solve the equation t^2 - 4 = 0 for t.

To find the roots, we can use the quadratic formula or factoring. In this case, factoring is easier since this is a simple equation: t^2 - 4 = (t - 2)(t + 2) The equation has two roots, t = 2 and t = -2.

Determine if the Wronskian is non-zero for any value of t.

We found that the Wronskian has two roots, t = 2 and t = -2. This means that the Wronskian is zero at these points. However, for any other value of t, the Wronskian will be non-zero (for example, for t = 0, W(0) = 0^2 - 4 = -4).

Determine if the functions are linearly independent or dependent.

Since the Wronskian is non-zero for at least one point in its domain (t = 0), the functions are linearly independent. So, we conclude that the functions are linearly independent.

Key concepts

linear independence

Linear independence is a concept that helps determine whether a set of functions or vectors are dependent on each other. Imagine you have a set of functions; they are linearly independent if none of them can be expressed as a combination of the others. In other words, no function in the set can be "made" using the others.
When dealing with the Wronskian, it serves as a handy tool for checking linear independence in the context of differential equations.

• If the Wronskian is non-zero for any point in the domain, the functions are linearly independent.
• If the Wronskian is zero everywhere, the functions can potentially be linearly dependent.

In the given exercise, the Wronskian was non-zero at points like \(t = 0\), indicating that the functions are indeed linearly independent.

differential equations

Differential equations involve unknown functions and their derivatives. They are equations that describe how functions change and are pivotal in modeling real-world phenomena.
The Wronskian pops up often when discussing differential equations, especially when determining the relationship between solutions.

• Solutions to differential equations can be expressed as a linear combination of functions.
• The Wronskian helps check if these solutions are linearly independent—important for understanding the behavior of solutions over time.

For instance, in our exercise, the Wronskian of two solutions being non-zero at least somewhere ensures these solutions are linearly independent, an important criterion in solving linear differential equations.

roots of equations

Finding the roots of an equation means solving for the values that make the equation equal to zero. For a quadratic equation, like in the exercise \(t^2 - 4 = 0\), this involves finding the values of \(t\) that satisfy the equation.
There are multiple ways to find roots:

• Factoring: breaking down the equation, as seen with \((t - 2)(t + 2) = 0\), yielding roots \(t = 2\) and \(t = -2\).
• Quadratic Formula: another reliable method, especially for more complex quadratics.

Knowing where the roots of the Wronskian are is crucial, as these points can hint at whether functions related to it are linearly dependent, which helps in understanding the nature of solutions to differential equations further. In the exercise, identifying the roots showed where the Wronskian becomes zero, helping deduce the linear independence of the functions involved.