Friday, July 26, 2013

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- 2y- 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 = òx (x /x3) dx + C = -1/x + C


So: y = -x2 + Cx3


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