Differential Equations and Slope Fields Intro

7/29/2019 Differential Equations and Slope Fields Intro

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Differential Equations and Slope Fields Name________________________Period:________

A differential equation is an equation that gives you information about how the slope (dy

dx) of

some function familyis related to the coordinates of points on the curve.

Example 1 The equation ydx

dy3= is a differential equation. The solution of this differential

equation will be a family of curves all satisfying the requirement that:At any point on one of the

curves, the slope (dy

dx) at that point will equal three times the y-coordinate of that point.

Evaluate: = )1,1(dx

dy

Explain in words___________________________________________________________________

How Are Differential Equations Solved?As with any equations, there are various methods of solving differential equations. We will

explore algebraic, geometric, and numerical ways to solve a diffeq.

Example 2 Given the differential equation yy = .

1. Evaluate the differential equation at the following points:

A. (1, -4) B. (3, 0) C. (2, )

2. Based on your answers to #1, what does this diffeq tell us about the nature of the curves

whose derivative satisfies yy = ?

3. Lets solve this differential equation geometrically.

At each coordinate, use the equation yy = to sketch

tiny segments with the appropriate slope. Any guessesas to what family of functions we have sketched?

4. Find the particular solution to this diffeq that passes through the point (0, 1) by usingthe tiny segments as little signposts directing your pencil.

A Separable Differential Equation will be in the form of ( ) ( )dy

g x h ydx

= . In order to solve it,

you must put it into the form of ( ) ( )j y dy g x dx= , allowing you to integrate each side

separatelyin terms of its only variable. Your goal is to get a final solution in the form ( ) .y f x=

Of course can also be written

ydx

dy=


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5. Now lets solve the differential equation from the previous page algebraicallyusing thetechnique called Separation of Variables. You will need to remember the catch-phrase

separate and integrate! To use this method, we will need to re-write y asdx

dy.

ydx

dy=

6. Find a particular solution to this diffeq by using the initial conditiony(0) = 1.

Example 3 Solve the differential equation subject to the given initial condition.

Pdt

dP3.0= ; 10)0( =P

Example 4 2004 AP

(a) Consider the differential equation ( )2 1dy

x ydx

= . On the axes provided, sketch a slope

field for the given differential equation at the 12 points indicated.

While the slope field in part (a) is drawn at only 12 points,it is defined at every point in the xy-plane. Describe allpoints in the xy-plane for which the slope is positive.

(b) Find the particular solution y=f(x) to the givendifferential equation with the initial condition f(0)=3.


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Example 5 Shown below is the slope field fordy

dx

x

y=

2

a. Calculatedy

dxat the points (3, 5) _____ and (-5, 1) _____

and demonstrate that the results agree with the figure.

b. Sketch the graph of the particular solution that containsthe point (5, 1). Draw on both sides of they-axis.

c. Solve the differential equation algebraically. Find theparticular solution that contains the point (5, 1).

Example 6 A sky diver jumps from an airplane. During the free-fall stage, the divers speedincreases at the acceleration of gravity, about 32.16 (ft/sec)/sec. But wind resistance causes aforce that reduces the acceleration. The resistance force is proportional to the square of the

velocity. Assume that the constant of proportionality is 0.0015, so thatdv

dtv= 32 16 0 0015 2. .

where v is in feet per second and t is in seconds. The slope field for this differential equationis shown here.

a) What does the slope appear to be at the point (5, 120)? _____

What does it actually equal? _____

Explain any discrepancy between your answers.

b) The diver starts at time t= 0 with zero initial velocity.Sketch the divers velocity as a function of time on the slope field.

c) The velocity in part (b) approaches an asymptote.What does this terminal velocity appear to be? ______

d) About how long does it take the diver to essentially reach this terminal velocity? _____

e) A second diver starts 5 seconds later with zero initial velocity. Sketch the velocity-timegraph.

f) Suppose the plane is going down steeply as a third diver jumps, giving that diver an initialdownward velocity of 180 ft/sec. Sketch this third divers velocity-time graph. How is itdifferent from the graphs you sketched for the first two sky divers? ______________

Consider the statement, The rate of change of some quantity y is directly proportional to y.


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Quantity = yrate of change ofy =

dy

dx

directly proportional =

multiplied by some constant k

So this statement can be translated: .dy

kydx

= kis sometimes called the growth (or decay) constant

Steps for solving:

dyk dt

y

= Separate the variables.

dyk dt

y=

Integrate both sides.

ln y kt C = + Perform the integration. Put the + Con the right side.

kt Cy e += Write this exponentially.

kt Cy e += Drop the absolute value as: e raised to any power is +.

Ckty e e= Split the right side into 2 factors.

kty Ae=ce is simply a constant, well callA.

When 0, .t y A= =

So, the statement: the rate of change of some quantityy is directly proportional toy is translated:

,

dyky

dt= which can be translated into .kty Ae=

KNOW THIS BACKWARD AND FORWARD!

Example 7 Punctured Tire Problem: You run over a nail. As air leaks out of your tire, the rate of change of air

pressure inside the tire is directly proportional to that pressure.

a) Write a differential equation that uses the fact: at

the time the nail is struck, the pressure is 35 lbs/psiand the pressure is decreasing at a rate of 0.28

lbs/psi/min.

b) Solve the differential equation.

c) What will the pressure be at 10 minutes after thetire was punctured?

d) The car is safe to drive as long as the pressure is 12lbs/psi or greater. For how long after the puncture will

the car be safe to drive?

Example 8 Dont Inhale Fumes: You accidentally inhale some poisonous fumes. Twenty hours later, you still

feel woozy so you go to a doctor. From blood samples, he measures a poisonous concentration of


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0.00372 mg/ml and tells you to come back in 8 hours. On the second visit, he measures a concentration of

0.00219 mg/ml.

Let t be the number of hours since your first doctor visit and Cbe the concentration of poison in your blood.

The rate of change ofCis directly proportional to the current value ofC.

a) Write a DEQ that relates these two variables. b) Solve the DEQ for the given data.

c) The doctor says that you might have serious bodily damage if the poison concentration has ever been as high

as 0.015 mg/ml. Based on your equation, was the concentration ever that high?

Differential Equations and Slope Fields Intro

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