Section Differential Equations: Modeling, Slope Fields, and Euler s Method

Transcription

1 Section.. Differential Equations: Modeling, Slope Fields, and Euler s Method Preliminar Eample. Phsical Situation Modeling Differential Equation An object is taken out of an oven and placed in a room where the temperature is 75 F. Let T(t) represent the temperature of the object, in F, after t minutes. dt = k(75 T), where k > is a constant. Description of the Phsical Law Modeled b the Differential Equation To the right, ou are given a slope field for the differential equation dt = k(75 T) for k =.5. For each initial condition shown below, draw the corresponding temperature function T(t) on the slope field. (i) T() = T (ii) T() = 5 (iii) T() = 5 (iv) T() = t Definition. A differential equation is an equation that involves an unknown function and one or more of its derivatives. Definition. A solution to a differential equation is a function that satisfies the differential equation. Definition. An initial value problem (IVP) is a differential equation together with one or more initial conditions. A solution to an IVP is a function that satisfies BOTH the differential equation AND the initial condition(s).

2 Eample. Let (t) represent the percentage of a particular task that has been learned after t months. Then can be modeled using the differential equation d = k( ), where k is a positive constant. (a) The slope field to the right represents the differential equation given above with k =.. Draw in the solutions to the following initial value problems: (i) d (ii) d =.( ), () = =.( ), () = 6 (b) Who learns faster at the beginning, a person who starts out knowing % of a task or a person who starts out knowing 6% of a task? Eplain (c) Eplain, in the contet of this situation, what the modeling differential equation d t = k( ) is saing. Eample. Let P(t) represent the population of a bacteria colon after t hours. Assume that this population is modeled b the differential equation dp =.P, whose slope field is given below (a) Draw in the solutions to the following initial value problems: (i) dp (ii) dp =.P, P() = =.P, P() = P 5 4 (b) Estimate the amount of time it takes for the population of a bacteria colon to double if it starts with bacteria. t 4 5 6

3 (c) For each of the following functions, use the slope field to guess whether the function could be a solution to the differential equation dp =.P. Then, use algebra and calculus to confirm our guesses. (i) P = e.t (iii) P = t + (v) P = (ii) P = e.t (iv) P = e.t+4 (vi) P = sint Eercises. Let k and C be an constants. Consider the differential equations given below: (I) d = (II) = For each of the functions given below, decide whether the function is a solution to (I), to (II), or to both (I) and (II). (a) = Ce t (b) = cost (c) = cost 4 sint (d) =

4 . (a) Show that all members of the famil = are solutions of the differential equation c =. (b) Use part (a) to find a formula for the solution to the initial value problem =, () =. Then, sketch our solution on the slope field shown to the right t Figure. Slope field for =. (a) Show that all members of the famil = + c are solutions of the differential equation = 5. (b) Find the solution to the initial value problem = 5, () = For what nonzero values of k does the function = e kt solve the differential equation 6 =? 5. (Taken from Stewart) A population is modeled b the differential equation dp ( =.P P ). 4 (a) For what values of P is the population increasing? (b) For what values of P is the population decreasing? 6. Below, ou are given 8 differential equations. Match each differential equation with the appropriate direction field (given on the net page). Then, complete parts (a) (c) below. (I) = (II) = (III) = (IV) = cos() (V) = (VI) = (VII) = (VIII) = (a) For each of the direction fields, draw in the solution that satisfies the initial condition () =. (b) Can ou guess a formula for an of the solutions ou drew in for part (a)? Check to see if ou re right! Definition. A constant solution to a differential equation is called an equilibrium solution. Graphicall, an equilibrium solution is a horizontal line, that is, a solution of the form = c for some constant c. 7. Use the above definition of equilibrium solution to answer the following: (a) Does the differential equation from the Preliminar Eample a few pages ago have an equilibrium solutions? If so, what are the? (b) Does the differential equation from Eample (preceding the eercises) have an equilibrium solutions? If so, what are the? (c) Does the differential equation from Eercise 5 above have an equilibrium solutions? If so, what are the? (d) Which of the differential equations from Eercise 6 above have an equilibrium solution? For those that do, what are the equilibrium solutions?

5 (A) (B) (C) (D) (E) (F) (G) (H)

6 Euler s Method Preliminar Eample. In the figure below, ou are given the slope field for an initial value problem of the form d = F(, ), () =. d Derive a method for approimating the solution curve () for this initial value problem Euler s Method Formulas: Eamples and Eercises. Consider the initial value problem d d =, () =. To the right, ou are given a slope field and a graph of the unknown solution to this problem, (). Use Euler s Method with step size.5 to estimate (.5). Sketch our solution curve on the slope field to the right and compare with the eact solution, ()

7 . The slope field for the differential equation = is shown to the right. Use Euler s method to approimate solution curves to the following initial value problems: (a) =, ( ) =.5.5 (b) =, ( ) =.5 Use a step size of.5 in both cases

8 Section.4 Separation of Variables Eample. (a) Solve d d =. Then, draw in several of our solution curves on the direction field given to the right. (b) Solve the initial value problem d d =, () =. Highlight this solution on our direction field above. (c) Solve the initial value problem d d =, () =. Highlight this solution on our direction field above.

9 Eample. For each of the following, ou are given a differential equation, a slope field, and an initial value problem. First, solve the differential equation and draw in several of our solution curves. Then, solve the initial value problem and highlight this particular solution curve. (i) d d = (ii) d d =, () = (i) d d = / (ii) d d = /, () =

10 (i) d d = e (ii) d d = e, () = Eamples and Eercises. Find the general solution to the differential equation + =.. Let v(t) represent the speed of a falling parachutist (before the chute opens), in feet per second, after t seconds. The motion of the parachutist is governed b the differential equation dv =.v. You ma assume that the parachutist has an initial speed of zero when jumping out of the plane; that is, v() =. (a) Find a formula for the speed of the parachutist as a function of time. (b) What happens to the speed of the parachutist as t? Use our answer from part (a) to justif our answer.. Solve e d = t +.