Consider the differential equation given by (dy/dx) = (xy/2)
a. sketch a slope field for the points (1,1), (1,2),(1,3), (0,1),(0,2), (0,3), and (-1,1), (-1,2), (-1,3)
b.let y=f(x) be the particular solution to the given differentialequation with the initial condition f(0)=3. Use Euler’s methodstarting at x=0 with step size of .1 to approximate f(0.2)
c.Find the particular solution to the differential equation y=f(x),given that (0,1) is a solution to the equation.
The Correct Answer and Explanation is :
Solution
Part (a): Sketching a Slope Field
The differential equation is:
[
\frac{dy}{dx} = \frac{xy}{2}.
]
To sketch the slope field, compute (\frac{dy}{dx}) for each given point ((x, y)). The slopes are:
- For ((1, 1): \frac{dy}{dx} = \frac{1 \cdot 1}{2} = 0.5),
- For ((1, 2): \frac{dy}{dx} = \frac{1 \cdot 2}{2} = 1),
- For ((1, 3): \frac{dy}{dx} = \frac{1 \cdot 3}{2} = 1.5),
- For ((0, 1): \frac{dy}{dx} = \frac{0 \cdot 1}{2} = 0),
- For ((0, 2): \frac{dy}{dx} = 0),
- For ((0, 3): \frac{dy}{dx} = 0),
- For ((-1, 1): \frac{dy}{dx} = \frac{-1 \cdot 1}{2} = -0.5),
- For ((-1, 2): \frac{dy}{dx} = -1),
- For ((-1, 3): \frac{dy}{dx} = -1.5).
Using these values, sketch the slope lines at each point to visualize the behavior of the differential equation.
Part (b): Euler’s Method
We approximate (f(0.2)) using Euler’s method with step size (h = 0.1) and initial condition (f(0) = 3). The formula for Euler’s method is:
[
y_{n+1} = y_n + h \cdot f'(x_n, y_n).
]
- Initial condition: (x_0 = 0), (y_0 = 3),
[
f'(x_0, y_0) = \frac{x_0 y_0}{2} = \frac{0 \cdot 3}{2} = 0.
]
(y_1 = 3 + 0.1 \cdot 0 = 3.) - Next step: (x_1 = 0.1), (y_1 = 3),
[
f'(x_1, y_1) = \frac{0.1 \cdot 3}{2} = 0.15.
]
(y_2 = 3 + 0.1 \cdot 0.15 = 3.015.)
Thus, (f(0.2) \approx 3.015.)
Part (c): Solving the Differential Equation
The equation is separable:
[
\frac{dy}{y} = \frac{x}{2} dx.
]
Integrate both sides:
[
\int \frac{1}{y} dy = \int \frac{x}{2} dx.
]
[
\ln|y| = \frac{x^2}{4} + C.
]
Exponentiate:
[
y = e^C e^{x^2/4}.
]
Let (e^C = C_1 > 0):
[
y = C_1 e^{x^2/4}.
]
Using the initial condition (f(0) = 1):
[
1 = C_1 e^{0^2/4} \implies C_1 = 1.
]
The particular solution is:
[
y = e^{x^2/4}.
]
Explanation
The differential equation (\frac{dy}{dx} = \frac{xy}{2}) describes a system where the rate of change of (y) depends on both (x) and (y). The slope field in Part (a) visualizes this behavior at discrete points. In Part (b), Euler’s method provides a numerical approximation for (f(0.2)), showing how the solution evolves from the initial condition. Finally, Part (c) derives the exact solution using separation of variables, confirming the general behavior of the system.