Solve the given differential equation by using an appropriate substitution.
(x-y)dx+ x dy=0

1 Answers

Best Answer

The differential equation is given as
(x-y)dx+xdy=0
Given differential equation can be written as
(dy)/(dx)=(y-x)/(x)=(y)/(x)-1
Let, y=vx. So, (dy)/(dx)=v+x(dv)/(dx)
Therefore, substituting y=vx in the given differential equation we have
v+x(dv)/(dx)=v-1
-> x(dv)/(dx)=-1
-> dv=-(dx)/(x)
-> int dv = - int (dx)/(x), (taking integration both side)
-> v=-ln x+c, (where c is an integrating constant)
-> (y)/(x)=-ln x+c, (putting v=(y)/(x))
-> y=x(c-ln x)
Result:
The solution of the given differential equation is y = x(c-ln x), where c is an integrating constant.