6.1
Slope Fields and Euler’s Method
Verifying Solutions
Determine whether the function is a solution of the
Differential equation y” - y = 0
a. y = sin x
y’ = cos x and y” = -sin x
y"-y = -sin x - sin x ! 0
So, y = sin x is not a solution.
b. y = 4e-x
y’ = -4e-x and y” = 4e-x
-x
-x
y"-y = 4e - 4e = 0
So, y = 4e-x is a solution.
y"-y = Ce x - Ce x = 0
So, y = Cex is a solution.
c. y = Cex
y’ = Cex and y” = Cex
Finding a particular solution
For the differential equation xy’ - 3y = 0, verify that
y = Cx3 is a solution, and find the particular solution
when x = -3 and y = 2.
y’ = 3Cx2
xy’ - 3y = x(3Cx2) - 3(Cx3) = 0
With the initial condition x = -3 and y = 2
y = Cx3
2=
C(-3)3
C=-
The particular solution is
2
27
2x 3
y =27
Sketching a Slope Field
Sketch a slope field for the differential equation
y’ = x - y for the points (-1,1), (0,1), and (1,1).
@ (-1,1) m = -1 - 1 = -2
@ (0,1) m = 0 - 1 = -1
@ (1,1) m = 1 - 1 = 0
-1
1
Sketch a slope field for the differential equation
y’ = 2x + y that passes through the point (1,1).
y|x -2 -1.5 -1 -.5
0
.5
-2 -6 -5 -4 -3
-2 -1
-1.5 -5.5 -4.5 -3.5 -2.5 -1.5 -.5
-1 -5 -4 -3 -2
-1 0
0
-4 -3 -2 -1
0
1
1
2
2
3
4
1
0
.5
1
2
1.5 2
1
2
1.5 2.5
2
3
3
4