Question

Let \(\left\\{\mathbf{y}_{1}(t), \mathbf{y}_{2}(t), \ldots, \mathbf{y}_{n}(t)\right\\}\) be a fundamental set of solutions of the linear system \(\mathbf{y}^{\prime}=P(t) \mathbf{y}\), where the matrix function \(P(t)\) is continuous on \(a

Short answer

Question: Prove that given a set of fundamental solutions of a linear system with continuous matrix function \(P(t)\), the set of solutions is linearly independent on the interval \((a, b)\). Answer: The set of solutions is linearly independent on the interval \((a, b)\) because the Wronskian of the set of solutions is nonzero on that interval.

Step-by-step solution

Rewrite the given equation

Write the given equation as \(\Psi(t) \mathbf{k} = \mathbf{0}\), where \(\Psi(t) = \left[\mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t)\right]\).

Choose an arbitrary point \(t_0\)

Choose an arbitrary point \(t_0\) in the interval \((a, b)\).

Evaluate \(\Psi(t_0)\)

Evaluate the matrix \(\Psi(t_0)\) by substituting \(t = t_0\) in the matrix function \(\Psi(t)\).

Analyze the equation \(\Psi(t_0) \mathbf{k} = \mathbf{0}\)

Now, we need to study the equation \(\Psi(t_0) \mathbf{k} = \mathbf{0}\). We want to show that the only solution is \(\mathbf{k} = \mathbf{0}\) because, if that is the case, it means that the set of functions \(\left\{\mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t)\right\}\) is linearly independent.

Apply the Wronskian

To prove that \(\mathbf{k} = \mathbf{0}\) is the only solution, let's compute the Wronskian of the given set of functions: \(W[\mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t)]\). If the Wronskian is nonzero, this implies that the set of functions is linearly independent.

Differentiate the Wronskian

Differentiate the Wronskian with respect to \(t\) and use the given equation \(\mathbf{y}^\prime = P(t) \mathbf{y}\) to show that the derivative of the Wronskian \(W(t)\) is equal to \(0\).

Evaluate the Wronskian at \(t_0\)

Evaluate the Wronskian at the arbitrary point \(t_0\). Since the derivative of the Wronskian is zero, the Wronskian is constant on the interval \((a, b)\). Hence, the Wronskian is nonzero, which means that the set of functions is linearly independent.

The conclusion

We have shown that the Wronskian of the set of solutions \(\left\{\mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t)\right\}\) is nonzero on the interval \((a, b)\). This means that the set of functions is linearly independent on the interval \((a, b)\).

Key concepts

Fundamental Set of Solutions

A fundamental set of solutions for a linear system of differential equations is a set of solutions that can represent any solution to the system through linear combinations. These solutions form a basis for the solution space of the differential equation. This concept is vital because it ensures that we can construct all possible solutions from a smaller, more manageable set.

• For a given linear system, such as \(\mathbf{y}' = P(t) \mathbf{y}\), finding a fundamental set of solutions is critical as it simplifies solving complex systems.
• The size of this set is determined by the order of the system, represented by the number of differential equations involved.

Understanding the composition of a fundamental set allows us to delve deeper into various properties of systems, including stability and response to inputs.

Linearly Independent Functions

Linearly independent functions are crucial in the context of differential equations because they describe a set of functions that do not simply scale one another. This independence means that there is no nontrivial linear combination of these functions that equals zero.

• In mathematical terms, if we have a set \(\{\mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t)\}\), they are linearly independent if \(c_1 \mathbf{y}_1(t) + c_2 \mathbf{y}_2(t) + \ldots + c_n \mathbf{y}_n(t) = \mathbf{0}\) implies that \(c_1 = c_2 = \ldots = c_n = 0\).
• This property ensures that each function contributes uniquely to the solution set.

Ensuring that a set is linearly independent is achieved, in part, by ensuring that the Wronskian is nonzero, which is a topic we will explore next.

Wronskian

The Wronskian is a determinant used to assess whether a set of functions are linearly independent over an interval. If the Wronskian of these functions is non-zero at any point within the interval, the functions are typically linearly independent on that interval.

• For functions \(\mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t)\), the Wronskian \(W\) is constructed from their derivatives, represented as a determinant in matrix form.
• If \(W eq 0\) at any point \(t_0\) in the interval, the functions are linearly independent. However, care must be taken, as the Wronskian being zero does not necessarily imply dependence.

The Wronskian plays a pivotal role in proving linear independence, which is important to understand the underlying behaviors of differential equation solutions.

Differential Equations

Differential equations are mathematical equations that involve an unknown function and its derivatives. They are widely used for modeling various systems in fields such as physics, engineering, biology, and economics.

• A linear differential equation takes the form \(\mathbf{y}' = A(t) \mathbf{y} + \mathbf{f}(t)\), where \(A(t)\) and \(\mathbf{f}(t)\) are functions of \(t\).
• Solving a differential equation involves finding the unknown function \(\mathbf{y}(t)\) that makes the equation true.
• Such equations can describe processes such as population dynamics, heat transfer, or electrical circuits.

Grasping the basics of differential equations opens up understanding for how systems evolve over time and interact within various applied contexts.