1. Part (a) dy dx =
1+y x dy dx =
1+1 1
m = 2
Keep in mind that dy/dx is the SLOPE! We simply need to substitute x and y into the differential equation and represent each answer as a slope on the graph.

2. Part (a) dy dx =
1+y x dy dx =
1+0 -2
m = -1/2
Repeating this process for the remaining six points gives us the following slope field…

3. Separate the variables
Part (b) dy dx =
1+y x
Cross multiply =
(1+y) dx
x dy
Separate the variables
Integrate both sides =
x-1 dx
(1+y)-1 dy =
ln x + C
ln 1+y

4. Plug in the initial condition
Part (b)
Plug in the initial condition =
ln C
ln 1+1
C = ln 2 =
ln x + ln 2
ln 1+y =
ln x + C
ln 1+y