LECTURE 3: SEPARABLE EQUATIONS
The form of a first order differential equation is F (x, y, y ′ ) = 0. An equation is said to be in
standard form if it can be written as follows:
dy
= f (x, y)
dx
There are some special cases of equations in standard form which have a straight-forward solution
strategy.
dy
(1) (Separable) dx
= g(x)h(y)
dy
= h(y)
(2) (Autonomous) dx
dy
(3) (Integrable) dx = g(x)
Certainly, the second two equations are both separable.
Example: Which of the following is separable?
dy
dy
dy
x+y
, dy
dx = ycos(x),
dx = e
dt = yt − t − y + 1,
dx = x + y
One can attempt to solve separable equations using the following very clever trick.
dy
= g(x)h(y)
dx
1 dy
= g(x)
=⇒
h(y) dx
Z
Z
1 dy
dx =
g(x) dx
=⇒
h(y) dx
Z
Z
1
dy =
g(x) dx
=⇒
h(y)
The last step is basically just u-substitution. So as long as we can do the integrals, we can eliminate
the differential and solve our differential equation.
Example: Solve the IVP using separation of variables: y ′ + y 2 sec2 (x) = 0, y(0) =
√
2.
Example: Solve the differential equation using separation of variables: y ′ = 1 − 2y.
Example: Solve the differential equation using separation of variables: y ′ = (1 + y 2 ) sin(x)/y
(leave in implicit form)
Example: Solve the logistic equation using separation of variables: P ′ = .25(4 − P )P .
1
1
4
=
+
Hint :
(4 − P )P
P
4−P
Example: Solve the differential equation y ′ = sin(y) (Don’t forget the straight line solutions!)
Example: A certain college math professor borrowed 11, 000 to buy a car. The bank charges
an annual interest rate of 10% a year. Suppose that Professor X makes annual payments of k dollars. Assume that both the annual interest and annual payment are continuous. How much should
Prof. X pay each year so that he pays off the car in three years? How much has he paid in interest
1
2
LECTURE 3: SEPARABLE EQUATIONS
at the end of the three years?
Example: (#38 from text) Suppose you are having a dinner party for a large group of people
and you decide to make 2 gallons of chili. The recipe calls for 2 teaspoons of hot sauce per gallon.
You misread the directions and put in 2 tablespoons) (1 tbsp =3 tsp =⇒ 12 tsp total in 2 gallons
of chili). You decide to serve it anyway, telling the people who like it hot to go first. As the guests
remove the chili, you add beans and tomatoes alone until the concentration is 2 tsp/gallon (from
the recipe). The guests are served at a rate of 1 cup/minute. Note there are 16 cups in a gallon.
How long will it take for the chili to get back to the right concentration?