Question

Let \(x^{(1)}, \ldots, x^{(n)}\) be linearly independent solutions of \(x^{\prime}=P(t) x,\) where \(P\) is continuous on \(\alpha

Short answer

Question: Show that any solution of the differential equation \(x' = P(t)x\) can be uniquely expressed as a linear combination of linearly independent solutions \(x^{(1)}, \ldots, x^{(n)}\), with constant coefficients. Answer: By Problem 11, Section 7.3, the Wronskian of the linearly independent solutions \(x^{(1)}, \ldots, x^{(n)}\) is nonzero. Using Problem 8, any solution can be expressed as a linear combination of these given solutions: \(z(t) = c_1x^{(1)}(t) + \cdots + c_n x^{(n)}(t)\). Assuming another set of constants \(k_1, \ldots, k_n\), we have \(0 = (k_1 - c_1) x^{(1)}(t) + \cdots + (k_n - c_n) x^{(n)}(t)\). The coefficients \((k_1 - c_1), \ldots, (k_n - c_n)\) must be zero, proving the representation is unique.

Step-by-step solution

a) Show that any solution z(t) can be expressed as a linear combination of linearly independent solutions

First, let \(z(t)\) be an arbitrary solution of the given differential equation, \(x' = P(t) x\). It is given that \(x^{(1)}, \ldots, x^{(n)}\) are linearly independent solutions of the equation. According to the result of Problem 11, Section 7.3, the Wronskian of the linearly independent solutions \(x^{(1)}, \ldots, x^{(n)}\) is given by a nonzero constant times an exponential function. Thus, the Wronskian \(W(t)\) does not vanish for any \(t \in [\alpha, \beta]\). Now, using Problem 8, we can write any solution \(z(t)\) as a linear combination of the given solutions \(x^{(1)}, \ldots, x^{(n)}\): $$ z(t) = c_1 x^{(1)}(t) + \cdots + c_n x^{(n)}(t) $$ for some constants \(c_1, \ldots, c_n\).

b) Showing the uniqueness of the expression for the solution z(t)

Assume that there exists another set of constants \(k_1, \ldots, k_n\) such that $$ z(t) = k_1 x^{(1)}(t) + \cdots + k_n x^{(n)}(t) $$ Subtracting these two expressions, we obtain: $$ 0 = (k_1 - c_1) x^{(1)}(t) + \cdots + (k_n - c_n) x^{(n)}(t) $$ Since it is given that \(x^{(1)}, \ldots, x^{(n)}\) are linearly independent solutions, and their Wronskian \(W(t)\) does not vanish for any \(t \in [\alpha, \beta]\), the only possibility is that each of the coefficients \((k_1 - c_1), \ldots, (k_n - c_n)\) is zero. Therefore, we have: $$ k_1 = c_1, \ldots, k_n = c_n $$ This shows that the representation of the solution \(z(t)\) as a linear combination of the given linearly independent solutions is unique.

Key concepts

Understanding Differential Equations

Differential equations are a type of mathematical equation that involve an unknown function and its derivatives. They are the backbone of dynamic systems and are used to describe how physical quantities change over time or under various conditions. For instance, they can represent the growth of populations, the oscillation of a pendulum, or the flow of electricity.

In our exercise, the differential equation given is of the first-order linear homogeneous type: \( x' = P(t) x \), where \( P(t) \) is continuous on an interval \( \alpha

Deciphering the Wronskian

The Wronskian is a specialized determinant used to establish the linear independence of a set of functions. Specifically, when considering a solution set of a differential equation, the Wronskian provides a criterion for deciding whether the solutions are linearly independent.

Think of it as a test: if the Wronskian of a set of functions is non-zero for some \( t \) within an interval, then those functions are linearly independent on that interval. In our exercise, the Wronskian of the solutions \( x^{(1)}, \ldots, x^{(n)} \) does not vanish for any \( t \) within \( [\alpha, \beta] \), which helps us confirm that these solutions are indeed linearly independent. Their independence is key to ensuring that the combination \( z(t) = c_1 x^{(1)}(t) + \cdots + c_n x^{(n)}(t) \) is unique, as no solution in the set can be expressed in the terms of others.

Applying the Superposition Principle

The superposition principle is a fundamental concept in linear systems, stating that the sum of individual solutions to a linear homogeneous differential equation is also a solution. This principle is at the heart of the exercise we’re examining.

Once we have a set of linearly independent solutions to a linear differential equation, any other solution can be expressed as a linear combination of these. That's what part (a) of the exercise demonstrates with the equation \( z(t) = c_1 x^{(1)}(t) + \cdots + c_n x^{(n)}(t) \). The constants \( c_1 \) to \( c_n \) are weights that adjust how much each independent solution contributes to the final solution \( z(t) \).

The exercise's part (b) leverages both the Wronskian and the superposition principle to prove that the coefficients \( c_1 \) to \( c_n \) in the expression for \( z(t) \) are unique. This is crucial because it not only provides a way to construct solutions from the known independent solutions but also ensures these solutions have a one-to-one correspondence—there’s no ambiguity in describing the system’s behavior.

Deciphering the Wronskian

The Wronskian is a specialized determinant used to establish the linear independence of a set of functions. Specifically, when considering a solution set of a differential equation, the Wronskian provides a criterion for deciding whether the solutions are linearly independent.

Think of it as a test: if the Wronskian of a set of functions is non-zero for some \( t \) within an interval, then those functions are linearly independent on that interval. In our exercise, the Wronskian of the solutions \( x^{(1)}, \ldots, x^{(n)} \) does not vanish for any \( t \) within \( [\alpha, \beta] \), which helps us confirm that these solutions are indeed linearly independent. Their independence is key to ensuring that the combination \( z(t) = c_1 x^{(1)}(t) + \cdots + c_n x^{(n)}(t) \) is unique, as no solution in the set can be expressed in the terms of others.

Applying the Superposition Principle

The superposition principle is a fundamental concept in linear systems, stating that the sum of individual solutions to a linear homogeneous differential equation is also a solution. This principle is at the heart of the exercise we’re examining.

Once we have a set of linearly independent solutions to a linear differential equation, any other solution can be expressed as a linear combination of these. That's what part (a) of the exercise demonstrates with the equation \( z(t) = c_1 x^{(1)}(t) + \cdots + c_n x^{(n)}(t) \). The constants \( c_1 \) to \( c_n \) are weights that adjust how much each independent solution contributes to the final solution \( z(t) \).

The exercise's part (b) leverages both the Wronskian and the superposition principle to prove that the coefficients \( c_1 \) to \( c_n \) in the expression for \( z(t) \) are unique. This is crucial because it not only provides a way to construct solutions from the known independent solutions but also ensures these solutions have a one-to-one correspondence—there’s no ambiguity in describing the system’s behavior.