Problem 19 - Calculus

Question

In \(19-22,\) find all solutions of the form \(y=e^{t}\). $$ y^{\prime \prime}+y=0 $$

Step-by-Step Solution

Answer

The solutions are in the form \( y(t) = c_1 \, \cos(t) + c_2 \, \sin(t) \).

Step 1: Assume the Solution Form

We are given that the solution is of the form \( y = e^{rt} \) where \( r \) is a constant. We need to determine \( r \).

Step 2: Calculate the First and Second Derivatives

Calculate the first derivative: \( y' = re^{rt} \), and the second derivative: \( y'' = r^2e^{rt} \).

Step 3: Substitute into the Differential Equation

Substitute \( y = e^{rt} \), \( y' = re^{rt} \), and \( y'' = r^2 e^{rt} \) in the equation \( y'' + y = 0 \). This gives us \( r^2 e^{rt} + e^{rt} = 0 \).

Step 4: Simplify the Equation

Factor out \( e^{rt} \) from \( r^2 e^{rt} + e^{rt} = 0 \) to obtain \( (r^2 + 1)e^{rt} = 0 \).

Step 5: Solve for the Characteristic Equation

Since \( e^{rt} eq 0 \), we have \( r^2 + 1 = 0 \). Solve for \( r \) to get \( r = \,i \) and \( r = -i \). These are the roots of the characteristic equation.

Step 6: Find the General Solution

With roots \( r = i \) and \( r = -i \), the general solution is \( y(t) = c_1 e^{it} + c_2 e^{-it} \). Using Euler's formula, this can be expressed as \( y(t) = c_1 \, \cos(t) + c_2 \, \sin(t) \).

Key Concepts

General Solution

In differential equations, finding the general solution is like discovering the complete behavior of the system you're analyzing. A differential equation is a mathematical equation that relates some function with its derivatives. To find the general solution, we solve the differential equation using appropriate techniques. In the example we have, the differential equation is given by \[ y'' + y = 0 \].

To tackle this, we look for solutions of the form \( y = e^{rt} \) where \( r \) is some constant. This form is used because it often simplifies the problem, particularly for linear differential equations with constant coefficients. By substituting \( y = e^{rt} \), and its derivatives back into the differential equation, we transform it into an algebraic equation involving \( r \). Solving this for \( r \), gives us the foundation to express the general solution.

The general solution accounts for all possible solutions of the differential equation. In our case, you would see that any combination of sine and cosine functions multiplied by constants \( c_1 \) and \( c_2 \) will satisfy our original equation, leading us to the general solution \( y(t) = c_1 \cos(t) + c_2 \sin(t) \).

Characteristic Equation

The characteristic equation is a pivotal concept for solving linear differential equations. After substituting the assumed form of \( y = e^{rt} \) into the differential equation \( y'' + y = 0 \), we derive the characteristic equation \( r^2 + 1 = 0 \).

This equation is derived based solely on the constants involved in the differential equation. Solving this characteristic equation provides us with the roots, which in this case are complex: \( r = i \) and \( r = -i \).

These roots signal that the solutions to our differential equation involve oscillatory components – very typical for differential equations resembling harmonic oscillators. Essentially, each root corresponds to an exponential function \( e^{rt} \) that makes up part of the complete solution. When these exponents are imaginary, as we see here, they imply sinusoidal functions when translated using Euler's Formula.

Euler's Formula

Euler's formula beautifully connects the world of exponential functions and trigonometry. It's stated as \( e^{ix} = \cos(x) + i\sin(x) \). This formula is immensely useful in simplifying and solving differential equations with complex roots.

In our example, the characteristic equation gave us complex roots \( r = i \) and \( r = -i \). By expressing the exponential solutions \( e^{it} \) and \( e^{-it} \) with Euler's formula, we reframe them in terms of sine and cosine functions: \( y(t) = c_1 \cos(t) + c_2 \sin(t) \).

Euler's formula is key because it transforms the complex exponentials into real-valued trigonometric functions. This transformation allows us to describe oscillating behaviors found in systems modelled by such differential equations. As a result, we gain a clearer and simpler form for the general solution, making it easier to interpret the real-world scenarios that the differential equation models.