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Magazine Chapter 5: Problem 16
Solve the given differential equation with initial condition. $$y^{\prime}-\frac{y}{7}=0, y(0)=6$$
Short Answer
Expert verified
y = 6e^{\frac{x}{7}}
Step by step solution
01
- Identify the Type of Differential Equation
Recognize that the given differential equation is a first-order linear differential equation of the form: \[ y' - \frac{y}{7} = 0 \]
02
- Rearrange the Equation
Rearrange the equation to isolate the derivative term on one side: \[ y' = \frac{y}{7} \]
03
- Solve by Separation of Variables
Separate the variables y and x: \[ \frac{dy}{y} = \frac{1}{7} dx \]
04
- Integrate Both Sides
Integrate both sides of the equation: \[ \begin{align*} \text{Left side:} & \ \ \ \text{Right side:} & \ \ \ \frac{dy}{y} & = \frac{1}{7} dx \ \ \rightarrow \ \ \text{Integrate:} & \ \ \ \ \text{Right:} & \ \ x \+_{0}^{'} = C \end{align*} \]
05
- Simplify and Solve for y
Simplify the result from the integration: \[ \text {ln}(\|land|{dy} communication ebetween(y) }function {}{};Subsect \tfield=equation subsection)_ entire C}^{ logarithm exponent} \]. Therefore: y= onumber C exponent^{0)\rightarrow{}}(\frac{x}{7 } }constants ; Subsection
06
- Apply Initial Condition
Use the given initial condition y(0) = 6 to find the constant C: \[ 6 = Ce^{0} \] So, \[ C = 6 \]. Therefore, the particular solution is:\[ y = 6e^{\frac{x}{7}} \]
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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 some function with its derivatives. In our exercise, the differential equation given is a first-order linear differential equation. Such equations involve the first derivative of the function and the function itself. Specifically, the equation given is:
\[y' - \frac{y}{7} = 0\]
This equation shows how the rate of change of y with respect to x (denoted as y') is related to y.
\[y' - \frac{y}{7} = 0\]
This equation shows how the rate of change of y with respect to x (denoted as y') is related to y.
Initial Condition
An initial condition is a value that is specified at the start of the problem so that a particular solution to a differential equation can be found. It allows us to determine the constant of integration that appears when we integrate during the process of solving the differential equation. In our exercise, the initial condition given is:
\[y(0) = 6\]
This means that when x is 0, y is equal to 6. We use this information to find the unique solution to our differential equation.
\[y(0) = 6\]
This means that when x is 0, y is equal to 6. We use this information to find the unique solution to our differential equation.
Separation of Variables
Separation of variables is a method to solve differential equations. The strategy is to rewrite the equation so that each variable appears on a different side. For our equation:
\[y' = \frac{y}{7}\]
We can separate the variables y and x by dividing both sides by y and multiplying by dx:
\[\frac{dy}{y} = \frac{1}{7} dx\]
This makes it possible to integrate each side independently.
\[y' = \frac{y}{7}\]
We can separate the variables y and x by dividing both sides by y and multiplying by dx:
\[\frac{dy}{y} = \frac{1}{7} dx\]
This makes it possible to integrate each side independently.
Integration
Integration is the process of finding the antiderivative, or the original function, from its derivative. After separating the variables in our differential equation, we integrate both sides to find:
\[\begin{align*}\text{Left side:} & \ \frac{dy}{y} & = \text{Right side:} & \ \frac{1}{7} dx \ \rightarrow \ \text{Integrate each side:} & \ \text{Left:} & \ \text{ln}\big| y \big| = \frac{x}{7} + C \ \text{Exponential:} & \ y = C e^{\frac{x}{7}}\end{align*}\]
We have used the properties of logarithms and exponents to simplify the results of our integration.
\[\begin{align*}\text{Left side:} & \ \frac{dy}{y} & = \text{Right side:} & \ \frac{1}{7} dx \ \rightarrow \ \text{Integrate each side:} & \ \text{Left:} & \ \text{ln}\big| y \big| = \frac{x}{7} + C \ \text{Exponential:} & \ y = C e^{\frac{x}{7}}\end{align*}\]
We have used the properties of logarithms and exponents to simplify the results of our integration.
Exponential Function
The exponential function appears frequently in the solutions of differential equations. In our problem, after integrating and simplifying, we obtain:
\[y = C e^{\frac{x}{7}}\]
The function \(e^{\frac{x}{7}}\) is the exponential function, where e is the base of the natural logarithm. This function grows rapidly as x increases. To find the constant C, we use the initial condition y(0) = 6:
\[6 = C e^{0}\]
Since \(e^{0} = 1\), this simplifies to:
\[C = 6\]
Plugging this value back in, we get the particular solution:
\[y = 6e^{\frac{x}{7}}\]
This solution describes the behavior of y with respect to x given the initial condition.
\[y = C e^{\frac{x}{7}}\]
The function \(e^{\frac{x}{7}}\) is the exponential function, where e is the base of the natural logarithm. This function grows rapidly as x increases. To find the constant C, we use the initial condition y(0) = 6:
\[6 = C e^{0}\]
Since \(e^{0} = 1\), this simplifies to:
\[C = 6\]
Plugging this value back in, we get the particular solution:
\[y = 6e^{\frac{x}{7}}\]
This solution describes the behavior of y with respect to x given the initial condition.
