Question

Consider the \(n\)th order differential equation $$ y^{(n)}-a y=0, $$ where \(a\) is a real number. In each exercise, some information is presented about the solutions of this equation. Use the given information to deduce both the order \(n(n \geq 1)\) of the differential equation and the value of the constant \(a\). (If more than one answer is \(|a|=2\) and all nonzero solutions of the differential equation are exponential functions.

Short answer

Question: Determine the order 'n' and the constant 'a' for a linear differential equation of the form \(y^{(n)}(t) - ay(t) = 0\), such that \(|a| = 2\) and all nonzero solutions are real exponential functions. Answer: The order of the differential equation is \(n = 1\) and the value of the constant is \(a = 2\).

Step-by-step solution

Exponential Function Formulation

Assume the solution is in the form of an exponential function \(y(t) = Ke^{rt}\). We will use this form to derive the \(n\)th order derivative.

Compute nth Order Derivative

The \(n\)th derivative of \( y(t) = Ke^{rt}\) is given by $$ y^{(n)}(t) = K\cdot r^n\cdot e^{r t}. $$

Plug in the solution to the Differential Equation

Substituting \(y(t)\) and \(y^{(n)}(t)\) into the given equation: $$ K\cdot r^n\cdot e^{r t} - aKe^{r t} = 0. $$

Factoring out Ke^{rt}

Factor out the common term, \(Ke^{rt}\): $$ Ke^{rt}(r^n - a) = 0. $$

Examine Nonzero Solutions

Since we are given that all nonzero solutions are exponential functions, \(Ke^{rt}\) is not equal to \(0\). Thus, we have: $$ r^n - a = 0 $$ which implies: $$ r^n = a. $$

Apply the given conditions

We know that \(|a|=2\). Thus, the value of \(a\) can be either \(2\) or \(-2\). The possible values of \(r^n\) will be subject to the order \(n\) and the values of \(r\) in the exponential function \(y(t) = Ke^{rt}\).

Determine the order n and the constant a

If \(a=2\), the equation becomes \(r^n = 2\). Since \(n \geq 1\), there will always be an exponential function. Thus, it satisfies the given condition. If \(a=-2\), the equation becomes \(r^n = -2\). However, for odd-order differential equations (\(n\) is odd), the exponential function \(y(t) = Ke^{rt}\) will have an imaginary component \(r\), which contradicts the given condition that all solutions are real exponential functions. Therefore, \(a\) cannot be equal to \(-2\). Thus, the equation is a first-order differential equation (\(n=1\)) with the constant \(a=2\): $$ y'(t)-2y(t)=0. $$

Key concepts

Exponential Functions

Exponential functions are mathematical functions of the form \( y(t) = Ke^{rt} \), where \( K \) and \( r \) are constants. These functions grow or decay at a rate proportional to their current value. In this context, the constant \( e \) is the base of natural logarithms, approximately 2.718. The rate \( r \) determines whether the function represents exponential growth (if \( r > 0 \)) or decay (if \( r < 0 \)).
Exponential functions are prevalent in solving differential equations because they can represent continuous growth or decay processes, like population growth or radioactive decay. In differential equations, the solutions often involve exponential functions because they naturally satisfy the characteristics of derivative relationships, as seen in the problem statement with solutions involving \( Ke^{rt} \).

• An essential property of exponential functions is that any derivative of an exponential function results in another exponential function, maintaining the same base \( e \), but adjusted by a factor related to the rate \( r \).
• Thus, the \( n \)th derivative of \( y(t) = Ke^{rt} \) becomes \( y^{(n)}(t) = K\cdot r^n\cdot e^{rt} \), which helps in solving higher-order differential equations, such as presented in the exercise.

Real Number Constant

A real number constant, like \( a \) in this differential equation, does not change regardless of the variable or parameter considered. Real numbers are values that represent a continuous quantity, which are both rational and irrational numbers.
In differential equations, constants like \( a \) are crucial as they impact the nature of the solutions. Specifically, the constant \( a \) here determines how the exponential function in the solution behaves. Given that \(|a|=2\), this means that \( a \) can either be \(-2\) or \(2\).

• These constants determine the fixed behavior of the system described by the differential equation. In this scenario, it affects whether the solution is strictly exponential (real and positive) or has a real and negative contribution, which could introduce complex elements if plugged into an exponential expression.
• Understanding that \( a \) operates as a real number constant allows the identification of valid solutions and constraints, which is crucial in distinguishing between possible solutions and ensuring they match given conditions.

Differential Equation Solutions

Differential equations like the one presented in this exercise are mathematical equations that relate a function with its derivatives. They describe how a particular quantity changes over time or space. The given equation \( y^{(n)}-a y=0 \) is a homogeneous linear differential equation of order \( n \), where \( n \) describes the highest derivative present.
The main goal of solving such an equation is to find a function (or a set of functions) that satisfies it for given conditions. In the context of exponential solutions:

• The form \( y(t) = Ke^{rt} \) is proposed, assuming that all non-zero solutions are exponential functions, implying the relationship \( r^n = a \).
• By substitution into the differential equation, and under given conditions, we determine relationships amongst the constants, namely that \( a \) must be \( 2 \) for the solution to exclusively involve real exponential functions.

Moreover, solutions to differential equations cover a variety of applications from simple physics problems to complex engineering systems. Solving them typically involves assessing both the order and the specific constants provided, just as was done when determining \( n = 1 \) and \( a = 2 \) in the provided exercise.