Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Which of the following is a solution to the differential equation $$ y^{\prime \prime}-y^{\prime}-6 y=0 ? $$ (a) \(y=C e^{t}\) (b) \(y=\sin 2 t\) (c) \(y=5 e^{3 t}+e^{-2 r}\) (d) \(y=e^{3 t}-2\)

Step-by-Step Solution

Answer

The correct solution is (c).

Step 1: Substitute the first option

Substitute \(y = Ce^t\) into the differential equation and see if the equality holds. This requires finding the first and second derivatives of \(y = Ce^t\).

Step 2: Substitute the second option

Substitute \(y = \sin 2t\) into the differential equation and see if the equality holds. This requires finding the first and second derivatives of \(y = \sin 2t\).

Step 3: Substitute the third option

Substitute \(y = 5e^{3t} + e^{-2t}\) into the differential equation and see if the equality holds. This requires finding the first and second derivatives of \(y = 5e^{3t} + e^{-2t}\).

Step 4: Substitute the fourth option

Substitute \(y = e^{3t} - 2\) into the differential equation and see if the equality holds. This requires finding the first and second derivatives of \(y = e^{3t} - 2\).

Step 5: Compare and draw conclusion

Compare the left-hand side and right-hand side for each option substitution. The function which makes the differential equation hold true is the correct answer.

Key Concepts

Homogeneous Differential Equations

A homogeneous differential equation is a type of differential equation in which the function and all its derivatives are expressed without any constant terms or non-zero numbers. In the given problem, the differential equation \(y'' - y' - 6y = 0\) is homogeneous because there are no constant terms—all terms involve the function \(y\) or its derivatives.

Homogeneous equations can often be solved by finding a characteristic equation. This involves substituting a trial solution of the form \(y = e^{rt}\) into the differential equation and solving for \(r\). The solutions to the characteristic equation will help construct the general solution to the differential equation.

• If you have real and distinct roots, the general solution is a combination of exponential functions.
• If roots are real and repeated, extra terms such as \(t\) multiplied by the exponential term are added.
• Complex roots introduce trigonometric functions into the solution.

Second-Order Differential Equations

Second-order differential equations involve the second derivative of the function, which is essential in defining the behavior of the system described by the differential equation. In the equation \(y'' - y' - 6y = 0\), \(y''\) denotes the second derivative and is a key term that makes this a second-order equation.

Second-order differentiable equations can model a variety of physical phenomena, like oscillations, waves, and systems undergoing acceleration or deceleration. The general form involves not just \(y\) and its first derivative \(y'\), but also \(y''\). These equations can often be linear, as in our example, meaning they can be expressed as a linear combination of the function and its derivatives.

• They are solved by finding characteristic equations or using integrative techniques.
• Understanding the nature of the roots of the characteristic equation helps to form the general solution.
• Analysis often involves checking how well candidate solutions fit, by substituting them back into the original equation.

Solutions of Differential Equations

Finding solutions to differential equations, such as \(y'' - y' - 6y = 0\), involves substituting different trial functions and checking which one satisfies the equation completely. In the given exercise, several possible solutions were provided.

• \(y = C e^t\): This form uses an exponential function; solving involves checking if the derived functions satisfy the original equation.
• \(y = \sin 2t\): Represents a trigonometric function which typically doesn't satisfy this kind of linear homogenous equation since the derivatives add additional terms.
• \(y = 5 e^{3t} + e^{-2t}\): Here, the combination of exponentials needs to satisfy both the structure of derivatives and the original equation.
• \(y = e^{3t} - 2\): This binds an exponential term minus a constant, posing a challenge since true solutions do not usually include isolated constants in a homogeneous equation.

The correct solution involves correctly substituting into the differential equation, deriving, and matching the terms. By the end of this substitution method, the solution that follows the equation without leaving any remainder or incorrect term is confirmed as the correct answer.