Problem 25 - Calculus Concepts

Question

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. \(\frac{d y}{d x}=k y\)

Step-by-Step Solution

Answer

The differential equation can be solved by separation of variables. The general solution is \( y = C e^{kx} \).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( \frac{dy}{dx} = ky \), where \( k \) is a constant. Recognize that this is a first-order linear differential equation. Such equations are often solved by separation of variables.

Step 2: Separate Variables

To separate variables, rewrite the equation to have all \( y \) terms on one side and all \( x \) terms on the other side. This becomes \( \frac{dy}{y} = k \, dx \). This separates variables and sets up the equation for integration.

Step 3: Integrate Both Sides

Integrate both sides of the equation. The left side gives \( \int \frac{1}{y} \, dy = \ln|y| \) and the right side gives \( \int k \, dx = kx + C \), where \( C \) is the constant of integration.

Step 4: Solve for y

Solve for \( y \) by exponentiating both sides to remove the logarithm. This gives \( |y| = e^{kx+C} = e^C e^{kx} \). Set \( e^C = C_1 \), a positive constant, giving \( y = C_1 e^{kx} \) or \( y = -C_1 e^{kx} \). Since \( C_1 \) can be positive or negative, the solution is \( y = C e^{kx} \), where \( C \) is an arbitrary constant.

Key Concepts

Antiderivatives

Antiderivatives, often referred to as indefinite integrals, represent a fundamental concept in calculus. When solving differential equations, finding the antiderivative allows us to reach a general solution. The antiderivative of a function gives us a way to reverse differentiation, providing the original function before it was differentiated. In other words, if we know a function's derivative, the antiderivative gives us a family of functions that could have been differentiated to obtain it.For example, if you have a function like \( f(x) = 2x \), its antiderivative could be \( F(x) = x^2 + C \), where \( C \) is the constant of integration. It's crucial to remember this constant because the derivative of a constant is zero, which means many different functions can have the same derivative. Using antiderivatives in the context of differential equations involves integrating both sides of an equation to find the original function from its derivative. This step is key in solving first-order differential equations.

Separation of Variables

Separation of variables is a powerful method used to solve differential equations. This technique is particularly useful for first-order differential equations. The basic idea is to rearrange the equation so that all terms containing one variable, typically \( y \), are on one side, and all terms containing the other variable, generally \( x \), are on the other side. For example, given the differential equation \( \frac{dy}{dx} = ky \), you would rearrange it to obtain \( \frac{dy}{y} = k \, dx \). Once separated, you can integrate both sides independently. This step is crucial because it transforms a complex equation into something more manageable. After integration, you obtain a relation between the variables, which often involves a logarithmic or exponential function because those are the kinds of functions involved in many real-world applications of differential equations. Finally, apply algebraic techniques to solve for one of the variables, typically \( y \), to find the general solution. The ultimate goal is to express the solution as an explicit function of one of the variables.

First-Order Linear Differential Equations

First-order linear differential equations are equations that involve the first derivative of a function. These equations take the general form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \). These equations are crucial because they model many natural phenomena, including exponential growth and decay, cooling laws, and even electrical circuits. Understanding them is key to applying mathematics to model the real world.To solve these equations, one common approach is separation of variables, which works well when the equation can be rearranged into the form \( \frac{dy}{dx} = ky \). Once separated, these problems are reduced to integrating both sides, a procedure that often involves simple logarithmic and exponential manipulation. Recognizing the structure of first-order linear differential equations allows you to approach problems systematically with confidence.