Question:
Consider the differential equation dy/dx = 2x - y.
(a) Create a slope field for this differential equation using the grid provided below. Label the x-axis from -2 to 2, and the y-axis from -2 to 2. Place at least five slope vectors on the grid.
| | | | |
___|___|___|___|___|
| | | | |
___|___|___|___|___|
| | | | |
___|___|___|___|___|
| | | | |
___|___|___|___|___|
| | | | |
(b) Use the slope field to sketch a particular solution to the given differential equation that passes through the point (1, 1).
(c) Use separation of variables to find the general solution to the differential equation.
(d) Use the initial condition y(0) = 3 to find the particular solution to the differential equation.
Answer:
(a) To create a slope field, we will calculate the slope at various points on the grid using the given differential equation. The slope at a point (x, y) is given by 2x - y. We will use the grid provided to place at least five slope vectors.
| | | | |
___|___|___|___|___|
| | | ← | |
___|___|_←_|___|___|
| |← |← |← |
___|___|___|← |___|
| | | | |
(b) Given the slope field, we can sketch a particular solution that passes through the point (1, 1). Starting from this point, we follow the slope vectors to get an approximation of the curve.
| | | | |
___|___|___|___|___|
| | | ← | |
___|_\←_|_←_|___|___|
|← |← |← |← |
___|___|___|___|___|
| | | | |
(c) To find the general solution to the differential equation, we will use separation of variables. Rearrange the equation:
dy/dx = 2x - y
dy = (2x - y)dx
dy + ydx = 2xdx
Divide both sides by y:
dy/y + dx = 2xdx / y
Integrate both sides:
∫(dy/y) + ∫dx = ∫(2xdx / y)
ln|y| + x = x^2 + C
Where C is the constant of integration.
(d) Using the initial condition y(0) = 3, we can find the particular solution. Substitute x = 0 and y = 3 into the general solution:
ln|3| + 0 = 0 + C
ln|3| = C
Therefore, the particular solution to the differential equation with the initial condition y(0) = 3 is:
ln|y| + x = x^2 + ln|3|