Problem 4 - Calculus Early Transcendentals

Question

State the order of the differential equation, and confirm that the functions in the given family are solutions. $$ \begin{array}{l}{\text { (a) } 2 \frac{d y}{d x}+y=x-1 ; \quad y=c e^{-x / 2}+x-3} \\ {\text { (b) } y^{\prime \prime}-y=0 ; \quad y=c_{1} e^{t}+c_{2} e^{-t}}\end{array} $$

Step-by-Step Solution

Answer

(a) Order 1, solution confirmed; (b) Order 2, solution confirmed.

Step 1: Determine the Order of the Differential Equation (a)

The given differential equation is \( 2 \frac{dy}{dx} + y = x - 1 \). The order of a differential equation is the highest derivative present in the equation. In this case, the highest order derivative is \( \frac{dy}{dx} \), which is the first derivative. Therefore, the order of this differential equation is 1.

Step 2: Confirm the Family of Solutions for Equation (a)

Given the solution \( y = c e^{-x / 2} + x - 3 \), we need to verify it satisfies the differential equation. First, calculate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = -\frac{c}{2} e^{-x / 2} + 1 \] Substitute \( \frac{dy}{dx} \) and \( y \) into the differential equation:\[ 2(-\frac{c}{2}e^{-x/2} + 1) + ce^{-x/2} + x - 3 = x - 1 \]Simplify the left-hand side:\[ -ce^{-x/2} + 2 + ce^{-x/2} + x - 3 = x - 1 \]Which simplifies to \( x - 1 = x - 1 \), confirming that the family function is a solution.

Step 3: Determine the Order of the Differential Equation (b)

The given differential equation is \( y'' - y = 0 \). The highest order derivative present is \( y'' \) which is the second derivative. Thus, the order of this differential equation is 2.

Step 4: Confirm the Family of Solutions for Equation (b)

The solution given is \( y = c_1 e^{t} + c_2 e^{-t} \). We need to verify it satisfies the differential equation. Compute the second derivative:\[ y' = c_1 e^{t} - c_2 e^{-t} \]\[ y'' = c_1 e^{t} + c_2 e^{-t} \]Substitute \( y, y'' \) into the differential equation:\[ (c_1 e^{t} + c_2 e^{-t}) - (c_1 e^{t} + c_2 e^{-t}) = 0 \]Which simplifies to \( 0 = 0 \), confirming that the family function is a solution.

Key Concepts

Order of Differential Equation

When dealing with differential equations, the "order" is a crucial concept. It tells us the highest derivative of a dependent variable with respect to an independent variable. In simplest terms, look for the derivative with the most number of dashes or a number indicating its progression.

For instance:

• In the equation \( 2 \frac{dy}{dx} + y = x - 1 \), the highest derivative is \( \frac{dy}{dx} \), making it a first-order differential equation.
• In the case of \( y'' - y = 0 \), \( y'' \) represents a second derivative, so it's a second-order differential equation.

Understanding the order helps when selecting the appropriate methods for solving these equations, providing insights into the complexity and characteristics of the solution sets.

Family of Solutions

A 'family of solutions' describes a set of functions that satisfy a differential equation. These solutions incorporate constants that offer flexibility reflecting the initial or boundary conditions that could apply.

• For the equation \( 2 \frac{dy}{dx} + y = x - 1 \), the solution \( y = c e^{-x/2} + x - 3 \) includes the constant \( c \), representing any function from the family, each of which satisfies the equation.
• Similarly, for the equation \( y'' - y = 0 \), the solutions \( y = c_1 e^{t} + c_2 e^{-t} \) contain constants \( c_1 \) and \( c_2 \), each representing a possible solution member of that family.

Recognizing these families is useful, as they show us how solutions change with different conditions applied and provide a comprehensive view of all possible solutions available for the equation given.

First Derivative

The first derivative represents the rate of change or slope of a function, giving critical information about its behavior. In simple terms, it tells us how the function value changes as the input changes.

In the differential equation \( 2 \frac{dy}{dx} + y = x - 1 \), \( \frac{dy}{dx} \) is the first derivative of \( y \) with respect to \( x \). This equation requires knowing this derivative to define the rate at which \( y \) changes relative to \( x \). To verify the solution \( y = c e^{-x/2} + x - 3 \), we compute:

• \( \frac{dy}{dx} = -\frac{c}{2} e^{-x/2} + 1 \), illustrating both the exponential decay and linear growth components in the solution.

Therefore, understanding and applying the first derivative is key in modeling the dynamic behavior of solutions in such equations.

Second Derivative

The second derivative offers insights into the concavity or curvature of a function. It's like asking, "Is the slope increasing or decreasing, and how fast?" This provides information on how the function is bending over a region, which is crucial for understanding complex motion or changes.

For example, in \( y'' - y = 0 \), \( y'' \) is the second derivative, indicating how the rate of change of the rate of change (acceleration) is behaving. To ascertain the solution, we compute:

• First, \( y' = c_1 e^{t} - c_2 e^{-t} \), the first derivative.
• Then, \( y'' = c_1 e^{t} + c_2 e^{-t} \), showing how the increments and decrements in the rate align perfectly to simplify back to zero when combined with \( y \).

Thus, the second derivative is essential in understanding and analyzing the deeper dynamic qualities of solutions to differential equations.