Find y = f(x) by solving the differential equation dy/dx = x-1 with the initial condition f(1) = 0

Answer 1

dy/dx = x-1

Separate variables to obtain dy = (x-1)dx

So, ∫dy = ∫(x-1)dx

y = (1/2)x² - x + C

Since f(1) = 0, y = 0 when x = 1.

Substituting, we have 0 = (1/2)(1)² - 1 + C. So, C = 1/2.

Therefore, y = (1/2)x² - x + 1/2