7.2. QUALITATIVE BEHAVIOR OF SOLUTIONS TO DES
401
1
2
3
-2
-1
1
2
3
t
y
This most recent figure, which is called a slope field for the differential equation, allows
us to sketch solutions just as we did in the preview activity. Here, we will begin with the
initial value y(0) = 1 and start sketching the solution by following the tangent line, as
shown in the next figure.
1
2
3
-2
-1
1
2
3
t
y
We then continue using this principle: whenever the solution passes through a point at
which a tangent line is drawn, that line is tangent to the solution. Doing so leads us to the
following sequence of images.
1
2
3
-2
-1
1
2
3
t
y
1
2
3
-2
-1
1
2
3
t
y
401
1
2
3
-2
-1
1
2
3
t
y
This most recent figure, which is called a slope field for the differential equation, allows
us to sketch solutions just as we did in the preview activity. Here, we will begin with the
initial value y(0) = 1 and start sketching the solution by following the tangent line, as
shown in the next figure.
1
2
3
-2
-1
1
2
3
t
y
We then continue using this principle: whenever the solution passes through a point at
which a tangent line is drawn, that line is tangent to the solution. Doing so leads us to the
following sequence of images.
1
2
3
-2
-1
1
2
3
t
y
1
2
3
-2
-1
1
2
3
t
y