Problem 21 - Differential Equations and Linear Algebra

Question

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=C^{2}(I),\) and \(S\) is the subset of \(V\) consisting o those functions satisfying the differential equation $$ y^{\prime \prime}+2 y^{\prime}-y=0 $$ on \(I\).

Step-by-Step Solution

Answer

The set S can be expressed in set notation as \( S = \{ y(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t} : c_1, c_2 \in \mathbb{R} \} \). After checking that the zero function is in S, and verifying that S is closed under addition and scalar multiplication, we can conclude that S is a subspace of the given vector space V.

Step 1: Find the general solution of the differential equation

To find the general solution of the given differential equation \(y^{\prime \prime} + 2y^{\prime} - y = 0\), we first find its characteristic equation: \[ r^2 + 2r - 1 = 0 \] Using the quadratic formula, we can find the roots of the characteristic equation: \[ r_{1,2} = \frac{-2 \pm \sqrt{2^2 - 4 \cdot (-1)}}{2} = -1 \pm \sqrt{2} \] Now that we have the roots, we can write down the general solution: \[ y(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t} \]

Step 2: Express S in set notation

Based on the general solution found in Step 1, S contains functions that satisfy the given differential equation. So, S can be expressed in set notation as follows: \[ S = \{ y(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t} : c_1, c_2 \in \mathbb{R} \} \]

Step 3: Check if the zero function is in S

The zero function, denoted as \(y_0(t) = 0\), is in S if there exist values of \(c_1\) and \(c_2\) such that the general solution equals zero. In this case, we can choose \(c_1 = 0\) and \(c_2 = 0\). Hence, the zero function is in S.

Step 4: Check if S is closed under addition

Let \(y_1(t)\) and \(y_2(t)\) be two functions in S: \[ y_1(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t} \] \[ y_2(t) = c_3 e^{(-1 + \sqrt{2})t} + c_4 e^{(-1 - \sqrt{2})t} \] To show that S is closed under addition, we need to first add \(y_1(t)\) and \(y_2(t)\) and then verify if their sum is also a function in S: \[ y_1(t) + y_2(t) = (c_1 + c_3)e^{(-1 + \sqrt{2})t} + (c_2 + c_4)e^{(-1 - \sqrt{2})t} \] The sum has the same form as the general solution, so S is closed under addition.

Step 5: Check if S is closed under scalar multiplication

Let \(y(t)\) be a function in S and k be any scalar: \[ y(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t} \] To show that S is closed under scalar multiplication, we need to first multiply \( y(t) \) by the scalar k and then verify if the product is also a function in S: \[ k \cdot y(t) = (k \cdot c_1) e^{(-1 + \sqrt{2})t} + (k \cdot c_2) e^{(-1 - \sqrt{2})t} \] The product has the same form as the general solution, so S is closed under scalar multiplication. Since S has the zero function, and it is closed under addition and scalar multiplication, we can conclude that S is a subspace of V.

Key Concepts

Differential Equations

In mathematics, differential equations are powerful tools that describe the relationship between a function and its derivatives. They come in many forms, varying from simple linear equations to complex non-linear ones.

The differential equation in our exercise, \(y'' + 2y' - y = 0\), is a homogeneous linear second-order differential equation. In general, solving such an equation involves finding a function \(y(t)\) that satisfies the equation for all values of \( t \).

Characteristics of Linear Differential Equations

Linear differential equations have consistent characteristics that define their solutions:

• They involve derivatives of a function.
• The function and its derivatives appear in a linear combination.
• In a homogeneous equation, the terms do not contain functions independent of the unknown function that's being solved for.

Understanding the solution's properties is critical for checking if the set of functions forms a subspace of a vector space, another fundamental concept in linear algebra.

General Solution

The general solution to a differential equation encompasses all possible solutions to the equation, usually represented through parameters that can take on various values.

For our specific exercise, the general solution to the given differential equation is expressed as \(y(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t}\), where \(c_1\) and \(c_2\) are arbitrary constants. A vibrant property of the general solution for linear differential equations is its ability to form a vector space by the principle of superposition.

Properties of the General Solution

The properties of the general solution that make it essential in our context include:

• It includes all particular solutions of the differential equation.
• The constants \(c_1\) and \(c_2\) can be chosen based on initial conditions or boundary values.
• It allows us to check whether a set of solutions forms a subspace by verifying the necessary conditions like closure under addition and scalar multiplication.

These aspects are crucial when identifying whether the solution set is a subspace of a larger vector space.

Set Notation

Set notation is a standardized format for describing collections of elements that share common properties.

In the context of our exercise, the solution set \( S \) can be defined using set notation to include all functions that are solutions to the differential equation. The notation
\[\ S = \{ y(t) = c_1 e^{(-1 + \sqrt{2})t} + c_2 e^{(-1 - \sqrt{2})t} : c_1, c_2 \in \mathbb{R} \} \]
delineates a set \( S \) containing functions of the form given by the general solution, where \(c_1\) and \(c_2\) are real numbers.

Importance of Set Notation in Subspaces

Set notation is essential in vector spaces and subspaces, as it precisely defines the elements being considered. Properties of sets that indicate a subspace include:

• Inclusion of the zero vector (or zero function, in the case of function spaces).
• Closure under addition and scalar multiplication, meaning that adding any two elements or scaling any element by a real number results in another element that is also in the set.

By using set notation, we can clearly identify the requirements for a subset to qualify as a subspace, strengthening our understanding of the structure and properties of vector spaces.