Slope Fields (6.1)
March 12th, 2013
I. General & Particular Solutions
A function y=f(x) is a solution to a differential
equation if the equation is true when y and its
derivatives are replaced by f(x) and its derivatives.
Ex. 1: Determine whether the function y=5lnx is a
(4)
solution of the differential equation y -16y=0.
You Try: Determine whether the function y=x2ex is a
3
x
solution of the differential equation xy’-2y=x e .
The general solution of a first-order differential
equation represents a family of solution curves that
include a constant C. Given initial conditions, we
can obtain a particular solution of the differential
equation.
Ex 2: Find the particular solution of the differential
equation 3x+2yy’=0 that satisfies the initial
condition y=3 when x=1, given that the general
solution is 3x2+2y2=C.
Ex. 3: Use integration to find a general solution of
dy
2
 x cos x
the differential equation
.
dx
You Try: Use integration to find a general solution
dy
 x/2

5e
of the differential equation
.
dx
II. Slope Fields
Slope fields, or direction fields, can be used to
analyze the solution to a differential equation that
cannot be found analytically. Given a differential
equation of the form y’=F(x, y) we can plug in any
point (x, y) to find the slope y’ of the solution of the
equation at that point. The sketch of short line
segments that show the slope at each point is
called a slope field.
Ex 4: Sketch
a
slope
field
for
the
differential
dy
equation dx  2  y
.
Ex. 5: (a) Sketch the slope field for the differential
equation y’=y+xy, (b) Use the slope field to sketch
the solution that passes through the point (0, 4),
and (c) discuss that graph of the solution as x  
and x  .