1. Repeating this process 8 more times gives us our slope field.
dy/dx =
y - 1 x2
Part (a)
(1/4)
(-1/4)
Let’s start with a random point—(2,2) for example...
dy/dx =
2 - 1 22
Repeating this process 8 more times gives us our slope field.
m = 1/4

2. Separate the variables
dy/dx =
y - 1 x2
Part (b)
(y-1)-1 = x-2 dx
Separate the variables
ln y-1 = -x-1 + C
Integrate both sides
ln = C
Find the value of C
0 = -1/2 + C
C = 1/2
ln y-1 = -x-1 + 1/2
( + ½) -1 x
ln y-1 e =
Exponentiate both sides

3. Part (b) ( + ½) e =  1 ( + ½) e = y-1 ( + ½) e = y-1  e0 =  1 1
( + ½) -1 2 e =  1
( + ½) -1 x e =
y-1
( + ½) -1 x e =
y-1  e0 =  1 1
Use the negative part.
( + ½) -1 x e = y  1
( + ½) -1 x e = y
1 -
( + ½) -1 x
ln y-1 e = e

4. Part (c) ( + ½) e 1 - lim = 1 – e0+½ = 1 – e½ 1 – e = ( + ½) e = y 1 -
( + ½) -1 x e
1 -
lim
x 
= 1 – e0+½
= 1 – e½
1 – e =
( + ½) -1 x e = y
1 -