Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 7-10, plot a slope field for each differential equation. Use the method of separation of variables (Section 4.9) or an integrating factor (Section 7.7) to find a particular solution of the differential equation that satisfies the given initial condition, and plot the particular solution. $$ y^{\prime}=\frac{1}{2} y ; y(0)=\frac{1}{2} $$

Step-by-Step Solution

Answer

The particular solution is \( y = \frac{1}{2} e^{\frac{1}{2}t} \).

Step 1: Understanding the Differential Equation

The given differential equation is \( y' = \frac{1}{2}y \). This is a first-order linear differential equation, where the rate of change of \( y \) is proportional to \( y \) itself.

Step 2: Setting up Separation of Variables

We want to use the separation of variables to solve the equation. We can rewrite the equation as \( \frac{dy}{y} = \frac{1}{2}dt \).

Step 3: Integrating Both Sides

Integrate both sides: \( \int \frac{1}{y} \, dy = \int \frac{1}{2} \, dt \). This results in \( \ln|y| = \frac{1}{2}t + C \), where \( C \) is the integration constant.

Step 4: Solving for y

Exponentiate both sides to solve for \( y \): \( y = e^{\frac{1}{2}t + C} = e^C e^{\frac{1}{2}t} \). Let \( A = e^C \) to simplify, so \( y = A e^{\frac{1}{2}t} \).

Step 5: Applying Initial Condition

Use the initial condition \( y(0) = \frac{1}{2} \) to find \( A \). Substitute into the equation: \( \frac{1}{2} = A e^{\frac{1}{2} \cdot 0} = A \). Hence, \( A = \frac{1}{2} \). The particular solution is \( y = \frac{1}{2} e^{\frac{1}{2}t} \).

Step 6: Plotting the Slope Field

To plot the slope field for the differential equation \( y' = \frac{1}{2}y \), compute the slope \( y' \) at several points, considering different values of \( y \) around each \( t \). The slopes will show lines that transform exponentially away from the \( t \)-axis.

Step 7: Plotting the Particular Solution

Plot the particular solution \( y = \frac{1}{2} e^{\frac{1}{2}t} \) on the same graph as the slope field. The curve should start at \( (0, \frac{1}{2}) \) and follow the slope lines from there, exponentially increasing as \( t \) increases.

Key Concepts

Slope Field

A slope field, also known as a direction field, is a visual representation in mathematics helping to understand differential equations. It consists of small line segments or arrows plotted on a plane. Each segment indicates the slope of the solution at that particular point.

• These slopes are derived from the differential equation in question.
• For example, in our exercise, the equation is \( y' = \frac{1}{2}y \), expressing how \( y \) changes with \( t \).
• At any point \((t, y)\), the slope \( y' \) is computed and plotted, showing how the solution evolves over time.

The slope field provides an intuitive guide to understanding the behavior of the differential equation's solutions without actually solving it analytically. By examining the direction field, one can surmise how solutions may behave—such as whether they grow or decay—as they progress through different points on the graph.

Separation of Variables

The separation of variables is a technique used to solve first-order differential equations. It is particularly useful when both sides of an equation can be expressed as a function of a single variable.
This method involves rearranging the differential equation so that each variable appears on separate sides of the equation.

• In our case, \( y' = \frac{1}{2}y \) can be rewritten
  as \( \frac{dy}{y} = \frac{1}{2} dt \).
• Notice how \( y \) and \( dy \) are on one side and \( t \) and \( dt \) are on the other side.

Once separated, you integrate each side with respect to its variable. This yields a solution involving an arbitrary constant, \( C \).
Separation of variables is often a preliminary step, leading to finding general solutions that form the groundwork for specific solutions given initial conditions.

Initial Condition

Initial conditions are crucial in determining particular solutions from the general solution of a differential equation.
Given the general solution from separation of variables, the initial condition allows us to pinpoint the specific solution that satisfies given circumstances.

• In our problem, the initial condition is given by \( y(0) = \frac{1}{2} \).
• This information is used to find the exact value of the arbitrary constant \( C \), which in turn determines the specific form of the solution.

Applying the initial condition, we solved for \( A \) in the equation \( y = A e^{\frac{1}{2}t} \) to find \( A = \frac{1}{2} \).
This step is important because without it, we would only have a general family of solutions rather than a specific, applicable one.

Particular Solution

A particular solution of a differential equation is one specific solution satisfying both the differential equation and any given initial conditions.
In the context of our exercise, we derived the particular solution from the general solution using the initial condition.

• The general solution from our separation of variables was \( y = A e^{\frac{1}{2}t} \).
• Using the initial condition \( y(0) = \frac{1}{2} \), we calculated \( A = \frac{1}{2} \).
• Thus, the particular solution became \( y = \frac{1}{2} e^{\frac{1}{2}t} \).

This particular solution is then plotted on the same graph as the slope field, starting from \((0, \frac{1}{2})\).
The curve follows the pattern outlined by the slope field, offering a clear visual representation of how solutions behave under specified conditions.