AP Calculus BC Unit 7 Differential Equations | Ohio State University

University:
Ohio State University
Course:
MATH 1151 | Calculus I
Academic year:

2021
Views:

476

Pages:

4
Author:

Yasmin Shaffer

Unit 7: Differential Equations ● Separation of variables: ○ When given the derivative equation and a function it’s equal to, you can find the original function using separation of variables. 𝑑𝑦 ○ Ex. find the equation for y in terms of x if 𝑑𝑥 = 2𝑥𝑦 and y(0) = 2 ■ 1) rearrange the equation so that each variable is separated from each other, so in this case, have all ‘y’ variables on one side, and all ‘x’ variables on the other → 𝑑𝑦 𝑦 = 2xdx ■ 2) integrate both sides of the integration → ∫ 𝑑𝑦 𝑦 = ∫ 2xdx = ln|𝑦|= 𝑥 2 + C ■ 3) solve for C by plugging in the values the question provides ( y(0) = 2) → ln|2|= C ■ 4) isolate y to find the equation of y → y = 𝑒 𝑥 2 +𝑙𝑛|2| ● Slope fields: graphical representation of a differential equation on a finite set of points in the plane ○ They basically tell you the derivative of a function at each point on a graph. The graph below is an example of a slope field. ■ Horizontal lines mean that the slope at that point is 0. ■ Vertical lines mean that the slope is undefined at the point. ○ How to sketch slope fields: ■ When asked to sketch the slope field of a function, you’re given the differential equation for that function. 𝑑𝑦 𝑥 ● Ex. 𝑑𝑥 = 𝑦 ○ All you do is plug in points (x,y) into the given differential equation to find the slope, and then make a small hash mark (just like the ones in the graph above) at that point to demonstrate the slope. ○ A slope field can be used to guess the equation of the original function ■ Many times, the shape of the slope field can kind of give an idea of the graph of the original function ● For example, looking at the slope field below (pictured on the left), you can see that the shape of the original function is most likely a variation of y=𝑥 3 (pictured on the right) ● Euler’s Method: ○ Used to find an approximate solutions using differential equations ○ Formula: , where ‘F’ = f’(x). ‘H’ = 𝛥𝑥 ○ Example (from Princeton book): Use Euler’s Method, with h = 0.2, to estimate y(1) if y’ = y - 2 and y(0) = 4 ○ *The important thing to remember with Euler’s formula is that the equation asks for the PREVIOUS x and y values ● Exponential growth: ○ Exponential growth and decay model is 𝑑𝑦 𝑑𝑡 = 𝑘𝑦 ■ With the initial condition y = y₀ when t = 0, the exponential growth model has solutions of the form y = 𝑦₀ 𝑒 𝑘𝑡 . ● Logistic growth: ○ This growth model is typically used with population ■ Carrying capacity refers to the maximum population that the environment can sustain; it’s the limiting value