8.1 Supplement: Differential Equations and Slope Fields
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1 Math 131 -copyright Angela Allen, Spring Supplement: Differential Equations and Slope Fields Note: Several of these examples come from your textbook Calculus Concepts: An Applied Approach to the Mathematics of Change, 4th ed., by LaTorre, Kenelly, Reed, Carpenter, Harris, and Biggers. Differential Equations: Equations that involve rates of change (derivatives) are called. Examples of differential equations include: the differential equation. for a differential equation is a function that has derivatives that satisfy Note: Finding a for a differential equation involves using a given to solve for the arbitrary constant, C. Note: is of the form dy = k (i.e. y = k), where k is a constant. is of the form dy = ax+b (i.e. y = ax+b).
2 Math 131 -copyright Angela Allen, Spring Example: (Example 1, pg. 509) Between 1900 and 2000, the population of the U.S. was growing at a rate of x million people per year, where x is the number of years since In 1990, the population of the U.S. was million people. a) Write a differential equation expressing the growth rate of the population with respect to time. b) Find a general solution for this differential equation. c) Find a particular solution for this differential equation. d) Estimate the population in the year Note: In order to use antiderivatives as solutions, the differential equation must satisfy the following: 1) The rate of change must be in terms of the input variable only. 2) We must know an antiderivative formula for the given rate of change function. Proportionality: Consider the constant differential equation of the form dy = k. This differential equation states that the rate of change of y with respect to x is constant. Consider the linear differential equation of the form dy = kx. This differential equation states that the rate of change of y with respect to x is to the input x.
3 Math 131 -copyright Angela Allen, Spring Direct Proportionality: For input x and output y, y is directly proportional to x if there exists some constant k such that y = kx. The constant k is called the. For instance, if A(t) = 23.50t dollars represents the amount it costs to purchase t tickets to a concert, then we say cost is directly proportional to the number of tickets purchased. Inverse Proportionality: For input x and output y, y is inversely proportional to x if there exists some constant of proportionality k such that y = k x. Example: (Example 2, pg. 511) Suppose that the total sales (in billions of dollars) of a computer product are growing in inverse proportion to ln(t + 1.2), where t is the number of years since the product was introduced. Sales totaled $53.2 billion by the end of the first year. a) Write a differential equation representing the rate of change of sales with respect to time. b) At the end of the first year, total sales were growing by 8.3 billion dollars per year. Find the constant of proportionality. c) Can we write an explicit formula for the general solution of this differential equation? Note: As we saw in the above example, we might not be able to find an explicit formula for the general solution of a differential equation. But, we can still analyze the differential equation (and it solutions) graphically by drawing.
4 Math 131 -copyright Angela Allen, Spring Slope Fields: slope field provides a graphical representation of a solution to a differential equation. slope field is constructed by placing a grid on a portion of the Cartesian plane and, at each point on the grid, drawing a short line segment whose slope is determined by the differential equation. Consider the differential equation dy = 2x. At the point (1,1), the slope of the solution (the quantity function of interest) is 2(1) = 2. So, at the point (1,1) we draw a short line segment with slope 2. At the point ( 0.5,2), the slope is 2( 0.5) = 1. So, at the point ( 0.5,2), we draw a short line segment with slope 1, and so on. The slope field for this differential equation is shown below: If we have an, we can sketch a particular solution for the differential equation on the slope field by following the line segments in such a way that the solution curves are tangent to each of the segments they meet. Example: Consider the differential equation dy = 2x. a) Find the particular solution to the differential equation with initial condition (0, 1). b) Use the slope field below to sketch the particular solution for the differential equation and initial condition given in part a above.
5 Math 131 -copyright Angela Allen, Spring Example: The slope field below shows three particular solutions for a differential equation. What are the three initial conditions each of these solutions corresponds to? Example: (Example 3, pg. 514) In a previous example, we derived a differential equation representing the rate of change ds of computer sales with respect to time: dt billion dollars per year, where t is the number of years after the ln(t + 1.2) product was introduced. a) Use the slope field of this differential equation shown below to sketch two particular solutions. b) Recall that we know sales were $53.2 billion at the end of the first year. Sketch the particular solution corresponding to this initial condition, and use it to estimate sales after 3 years. (Round to the nearest billion dollars.)
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