Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
WorkSheet 12
The Integrating Factor Technique
Step 1: Put the di¤erential equation in the form
dy
dx
+ f (x)y = g(x) (or
+ f (t)x = g(t))
dx
dt
dy
dy
= 3xy ¡ 4 as
¡ 3xy = ¡4 (so here f (x) = ¡3x and g(x) = ¡4)
dx
dx
dx
dx 2
Example: Write
= ¡t2 x + sin(t) as
+ t x = sin(t) (so here f (t) = t2 and g(t) = sin(t))
dt
dt
Step 2: Find the antiderivative of f (x): You do not need the \ + C”:
Example: Write
Step 3: If F (x) is the antiderivative of f(x); multiply both sides of the DE (in the proper
form) by eF (x) : (this function is called the integrating factor).
eF (x) (
dy
dy
+ f(x)) = g(x)eF (x) or eF (x)
+ f (x)eF (x) = g(x)eF (x)
dx
dx
³
Step 4: Notice that the left hand side is the derivative of a product. yeF (x)
´0
= g(x)eF (x)
Step
integrate
both sides of this new equation with respect to x:
Z
Z ³ 5: Now
´0
F (x)
ye
dx = g(x)eF (x) dx and notice that the left hand side is simply yeF (x)
Z
Final Step: Now solve for y: From Step 5, we have yeF (x) = g(x)eF (x) dx: After …nding the
antiderivative of the right hand side (Now remember to attach a “+C”), divide both sides
by eF (x) :
dy
= ¡3y + 4
Let’s go through the above steps with the easy example:
dx
Note that you could have used separation of variable in the previous example. Here’s an
dx
x
example where separation of variables won’t work:
=¡ +4
dt
t
Part of Homework Assignment due Monday, October 11
dy
= 2y + 4; y(0) = 3 in two di¤erent ways: separation of variables and integrating
1) Solve
dx
factor method.
2) Even the simple exponential growth DEs can be solved using the integrating factor method.
dx
Solve
= 3x; x(0) = 8 using the integrating factor method.
dt
3) Solve
dx
= ¡2x + sin(t)
dt
4) A 200 gallon tank initially contains 100 gallons of a solution of water and salt with the
initial amount of salt being 25 lbs. If pure water enters the tank at a rate of 6 gallons per
minute and the mixture leaves the tank at the rate of 4 gallons per minute, how much salt is
in the tank at the time the tank over‡ows? Use whatever method is appropriate for solving
this problem. Also give a graph of the solution.
5) A 200 gallon tank initially contains 100 gallons of a solution of water and salt with the
initial amount of salt being 25 lbs. If a brine solution containing 1/4 lb per gallon enters
the tank at a rate of 7 gallons per minute and the mixture leaves the tank at the rater
of 4 gallons per minute, how much salt is in the tank at the time the tank over‡ows? Use
whatever method is appropriate for solving this problem. Also give a graph of the solution.