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Chapter 3: Problem 1

Find the general form of \(f\) if \(f^{\prime}(x)=4 f(x)\).

Short Answer

Expert verified
The general form of \( f \) is \( f(x) = Ce^{4x} \).

Step by step solution

01

Understand the Differential Equation

The given equation is a first-order linear differential equation: \( f'(x) = 4f(x) \). It suggests that the rate of change of \( f(x) \) is proportional to its value.
02

Recall the Solution Form for This Type of Equation

For differential equations of the form \( f'(x) = kf(x) \), the general solution is \( f(x) = Ce^{kx} \), where \( C \) is a constant and \( k \) is a constant from the differential equation.
03

Apply the Solution Form to Our Equation

In our problem, the constant \( k \) is 4. Substitute \( k = 4 \) into the solution form to get: \( f(x) = Ce^{4x} \).
04

Confirm the General Solution

Verify that \( f(x) = Ce^{4x} \) satisfies the differential equation. Calculate the derivative: \( f'(x) = 4Ce^{4x} \), which equals \( 4f(x) \). This confirms that our solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Linear Differential Equations
First-order linear differential equations are a type of equation that involve the first derivative of a function. These equations have the general form \( f'(x) = a(x)f(x) + b(x) \), where \( a(x) \) and \( b(x) \) are given functions of \( x \).
This particular form highlights an important relationship: the rate of change of the function \( f(x) \) is linearly related to its value. In simpler terms, how fast the function is changing at any point is directly proportional to the function's current value.
  • In our example, the differential equation is \( f'(x) = 4f(x) \), meaning the rate of change is four times the value of the function.
  • These types of equations often appear in growth and decay problems, such as population growth or radioactive decay.
Understanding the structure of these equations is crucial, as it helps us find solutions easily by applying known methods for solving first-order linear equations.
Exponential Functions
Exponential functions are mathematical functions of the form \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants and \( e \) is the base of the natural logarithm, approximately equal to 2.718. These functions are powerful tools as they model phenomena where growth or decay accelerates over time.
Exponential functions have several key characteristics:
  • Their derivative is proportional to the function itself, meaning \( f'(x) = bf(x) \), which is evident in our problem.
  • They exhibit exponential growth when \( b > 0 \), like in our example where \( b = 4 \).
  • When \( b < 0 \), exponential decay is observed, common in scenarios like cooling processes or radioactive decay.
In our differential equation, recognizing that the solution takes the shape of an exponential function allows us to use derivatives to confirm that our solution satisfies the initial equation.
General Solution of Differential Equations
The general solution of a differential equation is a fundamental concept where we find a family of functions that satisfy the equation. It includes arbitrary constants that can be adjusted based on initial conditions.
For first-order linear differential equations of the form \( f'(x) = kf(x) \), the general solution format is \( f(x) = Ce^{kx} \), where \( C \) is an arbitrary constant.
  • In our case, \( k = 4 \), so the general solution is \( f(x) = Ce^{4x} \).
  • The constant \( C \) represents an integral constant that can be fixed if initial conditions are provided, tailoring the general solution to specific scenarios.
Ensuring the solution satisfies the differential equation is done by substituting and verifying that both sides of the equation match. This confirms that \( f(x) = Ce^{4x} \) is indeed a valid general solution to the problem at hand.

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Most popular questions from this chapter

In \(1970,\) the average salary of Major League baseball players was \(\$ 29,303 .\) In \(2013,\) the average salary was \(\$ 3,390,000 .\) Assuming exponential growth occurred, what was the growth rate to the nearest hundredth of a percent? What will the average salary be in \(2020 ?\) In 2025 ? Round your answers to the nearest thousand.

Spread by skin-to-skin contact or via shared towels or clothing, methicillin- resistant Staphylococcus aureus (MRSA) can easily infect growing numbers of students at a university. Left unchecked, the number of cases of MRSA on a university campus weeks after the first 9 cases occur can be modeled by $$ N(t)=\frac{568.803}{1+62.200 e^{-0.092 t}} $$ a) Find the number of infected students beyond the first 9 cases after 3 weeks, 40 weeks, and 80 weeks. b) Find the rate at which the disease is spreading after 20 weeks. c) Explain why an unrestricted growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation.

Differentiate. $$ y=\left(e^{3 x}+1\right)^{5} $$

Describe the differences in the graphs of an exponential function and a logistic function.

A beam of light enters a medium such as water or smoky air with initial intensity \(I_{0}\). Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity I at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$ The constant \(\mu\left({ }^{u} m u "\right),\) called the coefficient of absorption, varies with the medium. Use this law for Exercises 62 and \(63 .\) Concentrations of particulates in the air due to pollution reduce sunlight. In a smoggy area, \(\mu=0.01\) and \(x\) is the concentration of particulates measured in micrograms per cubic meter \(\left(\mathrm{mcg} / \mathrm{m}^{3}\right)\) What change is more significant-dropping pollution levels from \(100 \mathrm{mcg} / \mathrm{m}^{3}\) to \(90 \mathrm{mcg} / \mathrm{m}^{3}\) or dropping them from \(60 \mathrm{mcg} / \mathrm{m}^{3}\) to \(50 \mathrm{mcg} / \mathrm{m}^{3}\) ? Why?
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