1. First Order,
Separable
Differential
Equations

2. Differential Equation
mathematical equation for an unknown function
of one or several variables that relates the values
of the function itself and its derivatives of various
orders.

3. Let’s see some examples !
Solve the differential equation
dy
= -4xy 2
dx
and then solve the initial - value problem y (0) = 1
dy
= -4xy 2
dx
ò
1 dy
= -4x
2
y dx
1
dy = ò -4x dx
2
y
1
1=
2 × 02 - C
C = -1
1
dy = -4xdx
2
y
1
- = -2x 2 + C
y
y=
1
y= 2
2x - C
1
2x 2 +1

4. Practice Time!!!
Solve the initial – value problem:
dy
4y - cos y) - 3x 2 = 0
(
dx
4y - cos y) dy = 3x 2 dx
(
y ( 0) = 0
2 × 02 - sin0 = 03 + C
C=0
( 4y - cos y) dy = ò 3x 2 dx
ò
2y - sin y = x + C
2
3
2y2 - sin y = x 3

5. Slope Fields
Finding the solution given a differential
equation is often difficult.
We can come up with a visual pattern for
the general solution graphically using slope
fields.
• Let’s take the differential equation
y' = 2x - y
2
• Create a table showing several x, y, and
y’ values:

6. x
-2
-2
-1
-1
0
0
1
1
2
2
y
-1
1
-1
1
-1
1
-1
1
-1
1
y’
-5
-5
-3
-3
-1
-1
1
1
3
3
y' = 2x - y
2
Draw short segments
represent y’ at the
selected points.
We would need more
points to give an
accurate visual