Download Sections 6.3, 10.4, and Slope Field

Differential Equations
Def. A differential equation is an equation
that involves a function and some of its
derivatives.
Ex. y = 3y - 5y2
dy
5
 x  x , find the general solution.
Ex. Let
dx
A separable equation can be written
dy
 f  x g  y
dx
dy
To solve, we treat
as a fraction.
dx
Put the y’s on the left with dy, put x’s on
the right with dx, and integrate each side.
2
dP
t

Ex. Let
3 , find the general solution.
dt 2 P
dy
3
2
 x  sec x
Pract. 1)
dx
2
dR
t
2)

dt R
dy
2
3)
 12 y
dx
Def. An initial value problem (IVP) is a
differential equation and a value of the
solution.
dy 1
 if y(1) = 3.
Ex. Solve the IVP
dx x
dP
 3P if P(0) = 5.
Ex. Solve the IVP
dt
Important: A solution must be a function that
is differentiable over the largest interval
containing the initial value and satisfying the
original differential equation.
dy x
Ex. Find the solution to
if y (5) = -3

dx y
Slope Fields
Def. A slope field (or direction field) is a
diagram that shows the slope of a solution
at several points.
dy
Ex. Draw a slope field for
 x  y , then
dx
sketch a solution that satisfies y (0) = 0.
2
1
-2
2
-1
-2
Here’s what it would look like if we used
lots of points…