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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