Slope Fields & Differential Equations By Nathan, Hunter, & CJ
Slope Fields A slope field is a collection of small line segments with the same slope as the solution curve through a given point (x,y) Slope fields are made by plugging in values for an (x,y) coordinate into a differential equation, and using those values to solve for the slope at that point. The purpose for slope fields is to understand the direction that a graph is moving.
For example If dy/dx=3x+y, sketch a slope field with 9 points
Tips To stay organized, make a table of values When drawing a slope field by hand, you do not have to be perfect, but the grader needs to be able to see where your slope is smaller or greater. For multiple choice questions, check values that are far away from each other on the graph or have vertical/horizontal tangents, and eliminate answer choices as you go.
Differential Equations A differential equation is another way of writing a derivative. Rather than y’ or f’(x), a differential equation is represented by dy/dx. A separable differential equation is when you can isolate the variables, and get all of the y with dy, and all x with dx For example: dy/dx=(3x+2)/2y 2y*dy=(3x+2)dx This would allow you to eventually solve for f(x)
Particular Solutions in a Diff. EQ A particular solution to a differential equation is when it gives you a point the graph must pass through, which allows you to solve for the +c. For example, if dy/dx=2x/y, and f(3)=2, then f(x)=? Steps: Isolate the variables Integrate Plug in values for x and y Solve for c Note: Only the positive sq root works! Substitute value for c back into equation Solve for y
Particular Solutions in a Slope Field For a slope field, a particular solution allows you to take start at a point, and sketch along the slopes to approximate the function. For example, with the slope field for dy/dx=x^2, and the particular solution of f(1)=2, your particular solution could be approximated like this:
What is big C? Uppercase “C” is generally written as a replacement for e^c Used in differential equations where: dy/dx=ky, which after solving becomes: y=C^ekt You can write it the first way for an FRQ, but it becomes a much simpler problem using exponential properties, especially when further used for a particular solution.
Tips for Differential Equations Remember properties of exponents After solving for c, be careful to substitute the c value into the same step that x and y values were substituted. Read the question carefully