18.03 Class 4, Feb 15, 2006
First order linear equations: solutions.
[1]
form"
Definition: A "linear ODE" is one that can be put in the "standard
_____________________________



x' + p(t)x = q(t)

(*)
_____________________________
On Monday we looked at the Homogeneous case, q(t) = 0 :
x' + p(t) x = 0 .
This is separable, and the solution is xh = C e^{ integral p(t) dt}
Now for the general case.
Example: x' + kT = kT_ext, the (heat) diffusion equation.
k
is the "coupling constant." Let's take it to be 1/3.
(This cooler cost $16.95 at Target.)
Suppose the temperature outside is rising at a constant rate: say
T_ext = 60 + 6t (in hours after 10:00)
and we need an initial condition: x(0) = 32 .
So the equation is
x' + (1/3) x = 20 + 2t ,
x(0) = 32 .
This isn't separable: it's something new. We'll describe a method which
works for ANY first order liner ODE.
[2] Method: "variation of parameter," or "trial solution":
(1) First solve the "associated homogeneous equation"
x' + p(t) x
=
0
(*)_h
Write xh for a nonzero solution to it.
(2) Then make the substitution x = xh u , and solve for u .
(3) Finally, don't forget to substitute back in to get x .
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Staff
 Linear Equations, Equations, Constant of integration, xh, $16.95, nonzero solution

Click to edit the document details