AP Calculus BC4.1 Slope Fields & More Differential EquationsNov. 16/17, 2010
OPENER
Find the general solution (containing a constant C) to each Differential Equation (DE).
1) 2) 3)
4) Find the particular solution to the Initial Value Problem (IVP) at (1, 1)
LESSON
Since it can be difficult to find solutions to differential equations in an algebraic form, we can find approximate solutions graphically by drawing the slope field for a DE. Use the grid to draw the slope field of the given DE at each of the 36 integer coordinate points. That is, calculate dy/dx at each point (x, y) and draw a small line segment passing through each point which has the given slope value (dy/dx). For example, at the point (1, 1) the slope of the tangent line of the solution function is 1 and a line segment with slope 1 is drawn below.
1a)
b) Find the particular solution of the IVP at (0, 1) and graph the solution curve on the grid above.
c) Find the particular solution of the IVP at (0, -1) and graph the solution curve on the grid above.
Now let’s do more of these problems….but with the TI-nspire!
2) Using the Plot Differential Equation CAS document that I just upload on your nspire, plot the slope field of the given IVPsand move the initial point to see the corresponding changes in the solution function of the DE.
Recall: An Initial Value Problem (IVP) consists of a differential equation, and an initial condition (x, f(x)). The solution to an IVP is a function, y=f(x), whose graph passes through the initial condition. Thus, each point in a slope field has a solution curve passing through it.
a) b) c)
3) Now return to the DEs in this lesson’s OPENER and plot the slop field by choosing an initial point within a 3x3 viewing window centered at the origin. Check if the solution curve has the shape of the general solution you found.
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Newton’s Law of Cooling—A special IVP
“An object cools at a rate proportional to the temperature difference between the object and its environment.”
Let y(t) be the temperature of the object at time t and be the temperature of the environment. If the initial temperature of the object is y(0)= then Newton’s law of cooling is described by the following IVP:
The solution to this equation can be found either algebraically (by the separation of variables technique) or graphically by drawing a slope field and the solution curve passing through the point (0,).
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4) Show that is the general solution to the DE .
5) Two identical cups of dark liquid sitting in a 70°F laboratory cool according to Newton’s law. At t=0 the first cup’s temperature is 190°F and dropping at a rate of 12°F per minute. The second cup reaches 130°F after 10 minutes.
When did the first cup of liquid reach a temperature of 130°F?
Could the dark liquid in the second cup be coffee? Why or why not?
Start by writing an IVP for each liquid and finding the proportionality constant k.