Consider the differential equation given by dy/dx = xy/2. A. On the axes provided below, sketch a slope field for the given differential equation at the nine points indicated. B. Let y = f(x) be the particular solution to the given differential equation with the initial condition. Based on your slope field, how does the value of (0.2) compare to f(0)? Justify your answer. C. Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 3. Use your solution to find (0.2).
The Correct Answer and Explanation is :
Answer
Part A: Sketching the slope field
The given differential equation is:
[
\frac{dy}{dx} = \frac{xy}{2}.
]
To create the slope field, evaluate (\frac{dy}{dx}) at the nine specified points (e.g., ((x, y) \in {-1, 0, 1} \times {-1, 0, 1})):
- At ((0, y)): (\frac{dy}{dx} = 0), so the slope is 0 (horizontal lines).
- At ((x, 0)): (\frac{dy}{dx} = 0), so the slope is 0 (horizontal lines).
- At other points, calculate (\frac{dy}{dx} = \frac{xy}{2}) and sketch short line segments with the corresponding slopes.
For example:
- ((-1, 1)): (\frac{dy}{dx} = -\frac{1}{2}) (negative slope).
- ((1, 1)): (\frac{dy}{dx} = \frac{1}{2}) (positive slope).
- ((1, -1)): (\frac{dy}{dx} = -\frac{1}{2}) (negative slope).
Part B: Comparing (f(0.2)) and (f(0))
Given (f(0) = 3) and observing the slope field:
- Near (x = 0), the slopes are positive for (y > 0), indicating (y) increases slightly as (x) increases.
- Therefore, (f(0.2) > f(0)).
Part C: Finding the particular solution
The differential equation is separable:
[
\frac{dy}{y} = \frac{x}{2} dx.
]
Integrating both sides:
[
\ln|y| = \frac{x^2}{4} + C.
]
Simplify:
[
y = Ce^{\frac{x^2}{4}}.
]
Using the initial condition (f(0) = 3):
[
3 = Ce^{0}, \quad \text{so } C = 3.
]
The particular solution is:
[
y = 3e^{\frac{x^2}{4}}.
]
To find (f(0.2)):
[
f(0.2) = 3e^{\frac{(0.2)^2}{4}} = 3e^{0.01}.
]
Numerically:
[
f(0.2) \approx 3(1.01005) \approx 3.03015.
]
Explanation
- Slope Field: The slope field visualizes the behavior of solutions to the differential equation at specific points. Positive or negative slopes indicate increasing or decreasing solutions, respectively.
- Comparing Values: (f(0.2) > f(0)) because the slope field suggests that the solution grows as (x) increases near (x = 0).
- Solving the Equation: Using separation of variables and the initial condition, the solution is (y = 3e^{x^2/4}). Evaluating this at (x = 0.2) confirms (f(0.2) > f(0)).