Solutions to Differential Equations Using Integrating Factors

The problems again:

(1) Solve:  x²y dy – xy² dx – x³y²dx = 0

(2) Solve using any method for integrating factors:


x (dy/dx) - 3y = x²

Solutions:

(1)  Factor to obtain:: xy(xdy – ydx) – x² y²  dx = 0


Now, multiply by (x^{- 2}y^{- 2}):


(x dy – ydx)/ xy – x dx = 0

Then by applying the property of the differential:  d(ln y/x):

d(ln y/x) – xdx = 0

Integrating:: ln(y/x)  =   x²/2 +   c

Or:

y / x  =  c  exp (x²/2)  or:    y =   c x  exp (x²/2) 


(2) Put the equation into the form: dy/dx + Py = Q

Then: dy/dx – 3y/x = x

So: P = (-3/x) and Q = x


Therefore:

r = exp(ò Pdx) = exp (-3 ln x) = 1/ e ³lnx = 1 /x³

Whence:

(1/x³) y = ò_x (x /x³) dx + C = -1/x + C


So: y = -x² + Cx³