Find the equation of the general solution of the differentia

Find the equation of the general solution of the differential equation

xe^ydy/dx=4(x^6e^(4x)+e^y)

Your answer should be of the form y=f(x,C).

Solution

here the given equation is

(xe^y)dy/dx=4(x^6e^(4x)+e^y)

We observe that y occurs only in e^y , so we let v = e^y .

Then y = ln(v), so y\' =(1/v)v\' . Hence, the DE becomes

xv(1/v)v\'=4(x^6e^(4x)+v)

v\'-4(v/x)=4x^5e^(4x),   which is linear in v

The integrating factor is