Consider the differential equation dy/dx = x-1?

Question

1. Sketch a slope field for the given differential equation at the twelve indicated points.
2. Sketch the solution curve through the point (1, 0).
3. Find y = f(x) by solving the differential equation dy/dx = x-1 with the initial condition f(1) = 0

Answer 1

dy = (x-1) dx

Integrating...

y = (1/2)x^2 - x + c

Per initial condition: (1,0) ,

0 = (1/2)(1)^2 - 1 + c

0 = 1/2 - 1 + c

0 = -1/2 + c

c = 1/2

y = (1/2)x^2 - x + 1/2