IBDP. AI. HL. DEs. Slope Fields

Geometrically, we can interpret the differential equation $\frac{dy}{dx} = 2x + y$ as indicating the tangents of any solution curve of the equation. So we can find the gradient of the tangent at any point (x,y) on the solution curve by calculating the value of 2x + y at that point, eg. At point (1,-1) the tangent to the solution curve will have gradient 1.

The table below shows the gradients for each integer coordinate pair within $x,y \in [-2,2]$ :

Using short line segments to represent the gradient at each coordinate point gives a slope field of the tangents to the solution curves. Such a slope field is shown below, including three possible solution curves:

Worked Example 1

The slope field for $\frac{dy}{dx} = \frac{1 - x^2 - y^2}{ y - x + 2}$ is shown below:

a) Find the gradient of the tangent to the solution curves at (1,1)
b) Sketch the solution curves which passes through (1,1)

Worked Example 2

Consider the differential equation $\frac{dy}{dx} = x - y + 1$ , which has general solution $y = ce^{-x} + x$ .

a) Find the particular solution which passes through (0,-1)

b) Construct the slope field for the differential equation using the integer grid points for $x,y \in [-2,2]$ . Include the particular solution curve from part (a).

Exercise

1.) Use the slope field to graph the solution curve passing through (1,1):

2.) The slope field for the differential equation $\frac{dy}{dx} = \frac{-1 + x^2 + 4y^2}{y-5x+10}$ is shown below:

a) Find the gradient of the tangent to the solution curve at the origin
b) Sketch the particular solution passing through the origin.

3.) The slope field for $\frac{dy}{dx} = x^2$ is shown below:

a) Find the general solution to the differential equation
b) Find the particular solution for:
- (i) c = 1
- (ii) c = -2
Sketch these solutions on the slope field.

4.) The slope field for the differential equation $\frac{dy}{dx} = x(y-1)$ is shown below:

a) Sketch the solution curves which which passes through (0,2)
b) Find the equation of the solution curves drawn in part (a)

5.) The velocity v ms^-1 of a skydiver t seconds after she jumps from a helicopter satisfies the differential equation: $60 \frac{dv}{dt} = 60g - \frac{3g}{125} v^2$ , where $g \approx 9.8 ms^{-2}$ is the acceleration due to gravity. The slope field for this differential equation is shown below:

a) Given that the skydiver initially has velocity 0ms^-1, sketch the particular solution curve.
b) Describe what happens to the velocity of the skydiver as she falls.

6.) Consider the differential equation $\frac{dy}{dx} = 1 - x$ which has general solution $y = x - \frac{1}{2} x^2 + c$

a) Find the particular solution which passes through (-1,0)
b) Construct the slope field for the differential equation using the integer grid points for $x,y \in [-2,2]$ . Include the particular solution curve from part a.

7.) Consider the differential equation $\frac{dy}{dx} = \frac{1}{2} xy$

a) Find the particular solution which passes through (1,-1)
b) Construct the slope field for the differential equation using the integer grid points for $x,y \in [-2,2]$ . Include the particular solution from part a.

IBDP. AI. HL. DEs. Slope Fields

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