Need Calculus Math Help? The point P(7, −4) lies on the curve y = 4/(6 − x). If Q is the point (x, 4/(6 − x)), use your calculator to find the slope mPQ of the secant line PQ (correct to six decimal places) for the following values of x. Slope fields make use of this by imposing a grid of points evenly spaced across the Cartesian plane. At each point the value of the derivative is calculated and a short segment with that slope is drawn. These segments graphed together form the slope field. Here is the slope field for the differential equation.
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Calculus AB and Calculus BC
CHAPTER 9 Differential Equations
B. SLOPE FIELDS
In this section we solve differential equations by obtaining a slope field or calculator picture that approximates the general solution. We call the graph of a solution of a d.e. a solution curve.
The slope field of a d.e. is based on the fact that the d.e. can be interpreted as a statement about the slopes of its solution curves.
EXAMPLE 1
The d.e. tells us that at any point (x, y) on a solution curve the slope of the curve is equal to its y-coordinate. Since the d.e. says that y is a function whose derivative is also y, we know that
y = ex
is a solution. In fact, y = Cex is a solution of the d.e. for every constant C, since y′ = Cex = y.
The d.e. y′ = y says that, at any point where y = 1, say (0, 1) or (1, 1) or (5, 1), the slope of the solution curve is 1; at any point where y = 3, say (0, 3), (ln 3,3), or (π, 3), the slope equals 3; and so on.
In Figure N9–1a we see some small line segments of slope 1 at several points where y = 1, and some segments of slope 3 at several points where y = 3. In Figure N9–1b we see the curve of y = ex with slope segments drawn in as follows:
FIGURE N9–1a
FIGURE N9–1b
Figure N9–1c is the slope field for the d.e. Slopes at many points are represented by small segments of the tangents at those points. The small segments approximate the solution curves. If we start at any point in the slope field and move so that the slope segments are always tangent to our motion, we will trace a solution curve. The slope field, as mentioned above, closely approximates the family of solutions.
FIGURE N9–1c
EXAMPLE 2
The slope field for the d.e. is shown in Figure N9–2.
(a) Carefully draw the solution curve that passes through the point (1, 0.5).
(b) Find the general solution for the equation.
FIGURE N9–2
SOLUTIONS:
(a) In Figure N9–2a we started at the point (1, 0.5), then moved from segment to segment drawing the curve to which these segments were tangent. The particular solution curve shown is the member of the family of solution curves
y = ln x + C
that goes through the point (1, 0.5).
FIGURE N9–2a
FIGURE N9–2b
(b) Since we already know that, if then we are assured of having found the correct general solution in (a).
In Figure N9–2b we have drawn several particular solution curves of the given d.e. Note that the vertical distance between any pair of curves is constant.
EXAMPLE 3
Match each slope field in Figure N9–3 with the proper d.e. from the following set. Find the general solution for each d.e. The particular solution that goes through (0,0) has been sketched in.
(A) y′ = cos x
(B)
(C)
(D)
FIGURE N9–3a
FIGURE N9–3b
FIGURE N9–3c
FIGURE N9–3d
SOLUTIONS:
(A) goes with Figure N9–3c. The solution curves in the family y = sin x + C are quite obvious.
(B) goes with Figure N9–3a. The general solution is the family of parabolas y = x2 + C.
For (C) the slope field is shown in Figure N9–3b. The general solution is the family of cubics y = x3 − 3x + C.
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(D) goes with Figure N9–3d; the general solution is the family of lines y =
EXAMPLE 4
(a) Verify that relations of the form x2 + y2 = r2 are solutions of the d.e.
(b) Using the slope field in Figure N9–4 and your answer to (a), find the particular solution to the d.e. given in (a) that contains point (4, −3).
FIGURE N9–4
SOLUTIONS:
(a) By differentiating equation x2 + y2 = r2 implicitly, we get 2x + 2y from which which is the given d.e.
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(b) x2 + y2 = r2 describes circles centered at the origin. For initial point (4,−3), (4)2 + (−3)2 = 25. So x2 + y2 = 25. However, this is not the particular solution.
A particular solution must be differentiable on an interval containing the initial point. This circle is not differentiable at (−5,0) and (5,0). (The d.e. shows undefined when y = 0, and the slope field shows vertical tangents along the x-axis.) Hence, the particular solution includes only the semicircle in quadrants III and IV.
Solving x2 + y2 = 25 for y yields The particular solution through point (4,−3) is with domain −5 < x< 5.
Derivatives of Implicitly Defined Functions
In Examples 2 and 3 above, each d.e. was of the form = f (x) or y′ = f (x). We were able to find the general solution in each case very easily by finding the antiderivative
We now consider d.e.’s of the form where f (x,y) is an expression in x and y; that is, is an implicitly defined function. Example 4 illustrates such a differential equation. Here is another example.
EXAMPLE 5
Figure N9–5 shows the slope field for
At each point (x,y) the slope is the sum of its coordinates. Three particular solutions have been added, through the points
FIGURE N9–5
Slope Fields
Key Questions
7 4 Slope Field Sap Calculus Formulas
Example:
How do you draw the slope field for
#dy/dx = x - y# ?The slope field is a cartesian grid where you draw lines in various directions to represent the slopes of the tangents to the solution. Therefore by drawing a curve through consecutive slope lines, you can find a solution to the differential equation.
Take the example of
#dy/dx# at#(3, 4)# . Here we see that#dy/dx = 3 - 4 =-1# So you would draw a line of slope
#-1# at#(3,4)# . Repeat this for maybe 4 by 4 points to get the following slope fieldHopefully this helps!
If you are allowed to use a software, then a software called GeoGebra gives us the slope field below.
I hope that this was helpful.
Slope field can be used to visualize the flow of the solution curves of its corresponding differential equation. At each point of the solution curve, the curve will have the slope indicated in the slope field.