Question

For the following problems, find the general solution to the differential equation.\(y^{\prime}=4^{x}\)

Short answer

The general solution is \( y(x) = \frac{4^x}{\ln 4} + C \).

Step-by-step solution

Understand the Differential Equation

You are given a first-order differential equation of the form \( y' = 4^x \). This means the derivative of \( y \) with respect to \( x \) is equal to \( 4^x \). To find the general solution, we need to integrate with respect to \( x \).

Set up the Integral of Both Sides

The general solution requires integrating both sides. Therefore, integrate \( \int y' \, dx = \int 4^x \, dx \). This will give us the function \( y(x) \).

Integrate the Right-Hand Side

The integral of \( 4^x \) with respect to \( x \) can be expressed using the formula for exponential functions: \( \int a^x \, dx = \frac{a^x}{\ln a} + C \), where \( a \) is a constant. Here, \( a = 4 \), so we have \( \int 4^x \, dx = \frac{4^x}{\ln 4} + C \).

Write the General Solution

After integrating, we conclude that the general solution for the differential equation is: \( y(x) = \frac{4^x}{\ln 4} + C \), where \( C \) is an arbitrary constant.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In our problem, the expression \( 4^x \) is an exponential function. Here, 4 is the base and \( x \) is the exponent. These functions are important in mathematics because they model a variety of real-world phenomena such as population growth, radioactive decay, and interest calculations. Key features of exponential functions include:

• Growth: When the base is greater than one, the function models exponential growth.
• Decay: When the base is between zero and one, it models exponential decay.
• Continual Change: They are used to depict how quantities grow or shrink continuously over time.

Understanding these properties is crucial when working with differential equations, especially when the equations involve exponential terms like \( 4^x \). These equations require specific techniques, such as integration, to find solutions.

Integration Techniques

Integration is the process of finding a function, given its derivative. To solve our differential equation \( y' = 4^x \), we need to integrate both sides with respect to \( x \). This step is crucial in obtaining the function \( y(x) \) which is the solution to our differential equation.For the integration of exponential functions, we use:\[\int a^x \, dx = \frac{a^x}{\ln a} + C\]Here, \( a \) represents the base of the exponential function and \( C \) is the constant of integration. The natural logarithm \( \ln a \) appears in the denominator because derivative of \( a^x \) involves it through chain rule.Let's apply this to \( 4^x \):- Set up the integral: \( \int 4^x \, dx \)- Apply the formula: \( \int 4^x \, dx = \frac{4^x}{\ln 4} + C \)This integral gives us the function \( y(x) \) after integrating, which leads to understanding the differential equation's solutions comprehensively.

General Solution of Differential Equations

The general solution of a differential equation is a function that contains all possible solutions. For a first-order differential equation like \( y' = 4^x \), this involves integration and introducing a constant of integration denoted by \( C \). This constant represents the family of solutions, emphasizing that differential equations can have infinitely many solutions depending on initial conditions.Let's break it down:- **Differential Equation**: Our task starts with \( y' = 4^x \).- **Integrated Form**: After integrating \( 4^x \), we find \( y(x) = \frac{4^x}{\ln 4} + C \).- **Role of \( C \)**: The constant \( C \) signifies that solutions vary for different initial conditions such as \( y(x_0) = y_0 \).The expression \( y(x) = \frac{4^x}{\ln 4} + C \) encompasses all possible solutions determined by the differential equation. It is crucial to include \( C \) to represent the generality of the solution space. Understanding this concept is vital for fully capturing the behavior described by differential equations.