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Question

If the solution of differential equation x ln xdydx+y=2lnx is y(lnx)=(lnx)n+C then n = ___


Solution

Type of the differential equation is what you have to identify. Here if you divide the given equation by x lnx, it becomes dydx+1xlnx×y=2x which when compared to standard form of linear differential equation matches it. Here P(x) = 1xlnx
Q(x)=2x
Solution will be y X I.F = Q(I.F.)dx + C
I.F. = eP(x)dx
So I.F = lnx.
So general solution will be
y×lnx=2xlnxdx+C 
ylnx=(lnx)2+C 
So n = 2.

Mathematics

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