Differential equationsSolve the given differential equation by using an appropriate substitution.
2 months ago
Solve the given differential equation by using an appropriate substitution. (x-y)dx+ x dy=0
1 Answers
Best Answer
Browhywegottadothis Staff answered 2 months ago
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.
* For every student we do a unique answer