Slope Fields & Differential Equations By: Andrew Butterworth, John Wright, and Kailash Muthu

Differential Equations States how a rate of change in one variable is related to other variables Can be solved to find a function or class of functions Written in the form of dy/dx= derivative of a particular function

Solving Differential Equations Two Common Differential Equation Problems Given initial condition, solve for a particular solution: 1. Separate the variables such that all x’s are on one side and all y’s are on the other 2. Multiply the “dx” in dy/dx to the side with x’s 3. Integrate both sides, adding “+c” to the former “dx” side 4. Substitute the initial value into the integrated equations, solving for c 5. Substitute “+c” with the solved c value and solve for y

Example of Initial Condition Problem Course Description Exam Samples Calculus AB7 What is the particular solution to the differential equation dy/dx=4x/y where y(2)= -2? Separate the Variables Integrate both sides Add “+C to x side Solve for y using C value

Solving for Differential Equations Cont. 2nd Common Differential Equation Problem: solving for a general solution Used when lacking an initial condition 1.Split x’s and y’s through multiplication or division with dy on the “y’s” side and dx on the “x’s” side 2.Integrate both sides; don’t forget +c 3. Solve for general solution by substituting C for “+c” and solving for y

Example of General Solution Differential Equations 1993 BC13 appropriate for AB If dy/dx= x²y, what does y equal?

Special Case - Exponential Growth Function Initial value Rate Time Given that the rate is proportional to a certain value, dy/dt=ky and y=Ce^(kt) (or Ae^(kt))

Slope Fields A slope field is a graphical representation of the tangent lines on every point of a particular function. Helps with visualizing general functions and detecting patterns and trends. Ex: dy/dx 5 4 3 2 1 -1 -2 -3 -4 -5 Key notes: When dy/dx=0, there is a horizontal line on the slope field When the denominator of dy/dx=0, say x/0, a vertical line is typically written at that point

Common Strategies for Slope Field Questions Create a table relating all variables if necessary: x, y, and dy/dx, substituting values of x and y within the domain and range of the slope field I.E. dy/dx= -x/y x y dy/dx -1 1 1 (line pointing right on slope field) 0 (horizontal tangent) -1 (line pointing left on slope field) Undefined (possible vertical tangent)

Common Strategies for Slope Field Questions Cont. 2. Sketch/trace an image of the graph onto the slope field to determine the original function E.X.: dy/dx=x^2 Graph looks similar to x^3; thus, answer choice must be similar to x^3

Words of Advice When analyzing slope fields, try to plug in one point from each quadrant when trying to discover the equation the slope field is modeling. Don’t forget to write +- or |y| in your work in frq’s or points will be deducted. If you see a differential equation problem, you are most likely to be asked to find a particular solution. You need to be aware that such problem is usually 4 - 5 points and will have a massive impact on your FRQ score.

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