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Using an integration factor, Solve the following: (x+y)dx + (xlnx)dy = 0 Am I missing something simple here?
I came up with the integration factor of -x. As i multiply it through, it still does not seem to make the equation exact. Please help!!!
1 Answer
- ??????Lv 71 decade agoFavorite Answer
divide by x to put the equation in standard form :
dy/dx = (-x-y)/xlnx
dy/dx + (1/xlnx) y = -1/lnx
This is a linear first order diff. eq. that can be solved
using the normal procedure.
The standard form is y ' + p(x) y = g(x), so here we have
p(x) = (1/xlnx) and g(x)=-1/lnx
Then we have to calculate
m(x) = exp(integral(p(t) d(t)))
= exp(integral(d(lnt)/lnt))
= exp(ln(ln(x)) = ln(x)
y = (1/ln(x)) (integral( m(t) g(t) dt ) + C)
= (1/ln(x)) (integral( -1 dt ) + C)
= (1/ln(x)) (C - x) !