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Solving Linear Differential Equations of First Order

If the soluti...

Question

If the solution of differential equation $x l n \frac{x d y}{d x} + y = 2 l n x i s y (l n x) = (l n x)^{n} + C$ then n = ___

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Solution

Type of the differential equation is what you have to identify. Here if you divide the given equation by x lnx, it becomes $\frac{d y}{d x} + \frac{1}{x l n x} \times y = \frac{2}{x}$ which when compared to standard form of linear differential equation matches it. Here P(x) = $\frac{1}{x l n x}$
$Q (x) = \frac{2}{x}$
Solution will be y X I.F = $\int$ Q(I.F.)dx + C
I.F. = $e^{\int P (x) d x}$
So I.F = lnx.
So general solution will be
$y \times l n x = \int \frac{2}{x} l n x d x + C$
$y l n x = (l n x)^{2} + C$
So n = 2.

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