Math 152, Fall 2008
Lecture 13.
10/07/2008
The due date for HW#6 has been moved on Saturday, October
11, 11:55 PM.
Chapter 9. Further applications of integration
Section 9.1 Differential equations
Definition. Equation that contains some derivatives of an
unknown function is called a differential equation.
Definition. The order of a differential equation is the order of
the highest-order derivatives present in equation.
Definition. A function f is called a solution of a differential
equation if the equation is satisfied when y = f (x) and its
derivatives are substituted into the equation.
When we are asked to solve a differential equation we are expected
to find all possible solutions to this equation.
Now we are interested in first order differential equations
dy
= f (x, y )
dx
A separable equation is a first order differential equation that can
be written in the form
dy
= f (x)g (y )
dx
or
dy
f (x)
=
dx
h(y )
To solve this equation we rewrite it in the differential form
h(y )dy = f (x)dx
so that all x’s are on the one side of the equation and all y ’s are
on the other side. Then we integrate both sides of the equation
Z
Z
h(y )dy = f (x)dx
Then the general solution to this equation is
H(y ) = F (x) + C
where H(y ) is an antiderivative of h(y ), F (x) is an antiderivative
of f (x), C is a constant.
Example 1. Solve the equation
(a) y ′ =
ln x
xy + xy 3
(b) y ′ = 1 + y − x − xy
(c) y ′ =
x + sin x
3y 2
In many problems we need to find the particular solution to the
equation
dy
= f (x, y )
dx
that satisfies a condition
y (x0 ) = y0
The condition y (x0 ) = y0 is called the initial condition . The
problem of finding a solution to the differential equation that
satisfies the initial condition is called an initial value problem.
Example 2. Solve the initial value problem e y y ′ =
3x 2
,
1+y
y (2) = 0
Example 3. A brine solution of salt flows at a constant rate of
8L/min into a large tank tat initially held 100L of brine solution in
which was dissolved 0.5kg of salt. The solution inside the tank is
kept well stirred and flows out of the tank in the same rate. If the
concentration of salt in the brine entering the tank is 0.05kg /L,
determine the mass of salt in the tank after t min.
A first-order equation
dy
= f (x, y )
dx
specifies a slope at each point in the xy -plane where f is defined.
A plot of short line segments drawn at various points in the
xy -plane showing the slope of the solution curve there is called a
direction field for the differential equation. Because the direction
field gives the ”flow of solutions”, it facilitates the drawing of any
particular solution (such as the solution to an initial value
problem).
3
2
y(x)
1
K
3
K
2
K
0
1
1
2
x
K
1
K
2
K
3
Direction field for
dy
dx
= x2 − y
3
3
2
y(x)
1
K
3
K
2
K
0
1
1
2
x
K
1
K
2
K
3
Solutions to
dy
dx
= x2 − y
3
Section 9.2 First order linear equations
A first order linear differential equation is an equation that can be
written in the form
y ′ + P(x)y = Q(x)
where P(x) and Q(x) are continuous functions.
To solve this equation we have to find the integrating factor I (x)
such that
I (x)(y ′ + P(x)y ) = (I (x)y )′ = I (x)Q(x)
I (x) = e
R
P(x)dx
Then we have to solve the equation
d
[I (x)y ] = I (x)Q(x)
dx
So,
1
y (x) =
I (x)
Z
I (x)Q(x)dx + C
To solve the first order linear equation
(a) Find the integrating factor I (x) = e
R
P(x)dx
(b) Integrate the equation
d
[I (x)y ] = I (x)Q(x)
dx
and solve it for y by dividing by I (x).
Example 4. Find the general solution to the equation
ln x
y
.
y′ − = −
x
x
2
ex
y
,
Example 5. Solve the initial value problem y + 2 =
x
x
y (1) = e
′