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Magazine Chapter 1: Problem 7
Solve the initial value problem. $$ \frac{d y}{d x}=3 e^{x}, \quad y=6, \text { when } x=0 $$
Short Answer
Expert verified
The solution is \( y = 3e^x + 3 \).
Step by step solution
01
Identify the Differential Equation
We are given the differential equation \( \frac{d y}{d x} = 3 e^x \). This tells us how \( y \) changes with respect to \( x \).
02
Integrate Both Sides
To solve for \( y \), integrate both sides with respect to \( x \). The left side becomes \( y = \int 3 e^x \, dx \).
03
Perform the Integration
The right side \( \int 3 e^x \, dx \) integrates to \( 3e^x + C \), where \( C \) is the constant of integration. Thus, \( y = 3e^x + C \).
04
Apply the Initial Condition
We are given that \( y = 6 \) when \( x = 0 \). Substitute these values into the equation: \( 6 = 3e^0 + C \). Since \( e^0 = 1 \), this simplifies to \( 6 = 3 + C \). Solve for \( C \) to get \( C = 3 \).
05
Write the Solution
Substitute \( C = 3 \) into the integrated function to get the particular solution. Thus, \( y = 3e^x + 3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differential Equations
A differential equation is a mathematical equation that relates a function with its derivatives. In the context of this exercise, the differential equation is \( \frac{d y}{d x} = 3 e^x \). This equation presents a scenario where the rate of change of the function \( y \) with respect to \( x \) is equal to \( 3 e^x \). Understanding differential equations is crucial as they describe various real-world systems in physics, engineering, and biology.
Here are some key aspects of differential equations:
Here are some key aspects of differential equations:
- Order: The order of a differential equation is determined by the highest derivative present. In this case, it's a first-order differential equation.
- Linear or Non-linear: Our given equation is linear because it can be expressed as a linear combination of its variables and their derivatives.
Integration
Integration is the mathematical process of finding a function when its derivative is known. It essentially reverses differentiation.
In this problem, we have to integrate \( 3 e^x \) to find the function \( y \). When we integrate, we gather the 'pieces' accumulated by differentiation.
Here are some important points to remember about integration:
In this problem, we have to integrate \( 3 e^x \) to find the function \( y \). When we integrate, we gather the 'pieces' accumulated by differentiation.
Here are some important points to remember about integration:
- Indefinite Integration: When you perform integration without limits, you introduce an arbitrary constant \( C \). This is known as indefinite integration.
- Basic integration rule for exponential functions: When integrating \( e^x \), you get \( e^x \), and any constant multiple remains (e.g., \( \int 3e^x \, dx = 3e^x + C \))
Constant of Integration
The constant of integration, represented as \( C \), is an important part of the solution to a differential equation. It represents the unknown that arises when integrating a function.
When you integrate \( 3 e^x \), the result is \( 3e^x + C \). Because every derivative of a constant is zero, the exact value of \( C \) is not known just from the integration process itself. This is where initial conditions come into play.
When you integrate \( 3 e^x \), the result is \( 3e^x + C \). Because every derivative of a constant is zero, the exact value of \( C \) is not known just from the integration process itself. This is where initial conditions come into play.
- Determining \( C \): Initial conditions such as \( y = 6 \) when \( x = 0 \) help you find the value of \( C \). In this problem, substituting these known values into the integrated function allows you to solve for \( C \) and personalize the solution to the specific problem context.
Exponential Function
The exponential function \( e^x \) is a crucial mathematical function characterized by a constant rate of growth. In the context of differential equations, it often describes processes involving growth and decay.
For this specific problem, \( e^x \) appears as part of the solution to the differential equation \( \frac{d y}{d x} = 3 e^x \).
Key features of the exponential function include:
For this specific problem, \( e^x \) appears as part of the solution to the differential equation \( \frac{d y}{d x} = 3 e^x \).
Key features of the exponential function include:
- Base: The base \( e \) is an irrational number approximately equal to 2.71828.
- Derivative and Integral: The unique property of exponential functions is that the derivative of \( e^x \) is itself, and similarly, the integral of \( e^x \) is also \( e^x \) (plus a constant).
- Applications: Exponential functions model various real-world phenomena such as population growth, radioactive decay, and continuous compound interest. Their constant relative growth rate makes them essential in modeling behavior that changes exponentially over time.
