dy/dx = 6 - y Solve the differential equation

 
Apply direct integration both sides: intN(y) dy= int M(x) dx to solve for the  general solution of a differential equation.
 For the given first order ODE: (dy)/(dx)=6-y  it can be rearrange by cross-multiplication into:
(dy)/(6-y)=dx
Apply direct integration on both sides: int(dy)/(6-y)=int dx
 For the left side, we consider u-substitution by letting:
u=6-y then  du = -dy    or   -du=dy
The integral becomes: int(dy)/(6-y)=int(-du)/(u)
 Applying basic integration formula for logarithm:
int(-du)/(u)= -ln|u|
 Plug-in u = 6-y on   -ln|u| , we get:
int(dy)/(6-y)=-ln|6-y|
For the right side, we apply the basic integration: int dx= x+C
  
Combing the results from both sides, we get the general solution of the differential equation as:
-ln|6-y|= x+C
y =6-e^((-x-C))
 or 
y = 6-Ce^(-x)