Section 1.8. LINEAR EQUATIONS. Introduction. First, take a few minutes to read the introduction on p. 112. What is the only technique we have learned so far for actually solving DEs? Why aren’t you going to learn how to solve every DE in this course? (Is Dr. Hall being lazy?)
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First, take a few minutes to read the introduction on p. 112.
A first-order* differential equation is linear if it can be written in the form
where a(t) and b(t) are any functions of t.
*First order means that the equation contains only
first derivatives (dy/dt, not d2y/dt2).
A23, B4, C1, D13, E2
Suppose yh(t) is a solution of the homogeneous DE
and suppose yp(t) is a particular solution of the nonhomogeneous DE
Let y(t) = yh(t)+yp(t). Show that y(t) is a solution of the nonhomogeneous DE above.
Hint: plug y(t) into the nonhomogeneous equation.
A tale of two equations:
which is a nonhomogeneous DE (assume b(t)≠0), and
which is its associated homogeneous DE.
Make a hypothesis about the relationship between solutions of (NDE) and (AHDE). Check your hypothesis by plugging into (NDE).
Try to make your hypothesis as general as possible.
Hint: use the linearity principle if you haven’t already.
We haven’t developed any general techniques for finding even one solution of a nonhomogeneous DE.
Maybe we could just, like, guess a solution. Maybe we’d get lucky! After all, we only need one.
This extremely precise scientific technique is known as the METHOD OF LUCKY GUESS.
(To tell the truth, a lot of math and science works this way: you have a hunch that something is true, then you try to prove (or disprove) it.)
Read p. 118.
Read p. 119
qualitatively. What is the long-term behavior of every solution?