Step-by-step explanation:
a. separate the variables
[tex]\frac{\mathrm{d} y}{\mathrm{d} x}[/tex] = 500-y
dy/(500-y) = dx
b. integrating your equation in part a to find the general equation of
differential
Integrating on both sides
[tex]\int[/tex]dy/(500-y) = [tex]\int[/tex]dx
-㏑(500-y) = x +C ..............(1)
where C is constant of integration
c. If y(0) = 7
putting in equation (1)
-㏑(500-7) = 0+C
C = -㏑493
d. The particular solution is
-㏑(500-y) = x -㏑473
㏑473/(500-y) = x
473 = (500-y)[tex]e^x[/tex]
