Question

The value of  'n', such that the differential equation `x^ndy/dx = y(logy - logx + 1)`; (where x, y ∈ R⁺) is homogeneous, is ______.

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Answer 1

The value of  'n', such that the differential equation `x^ndy/dx = y(logy - logx + 1)`; (where x, y ∈ R⁺) is homogeneous, is 1.

Explanation:

A differential equation of the form `dy/dx = f(x, y)` is said to be homogeneous if f(x, y) is a homogeneous function of degree 0.

Now, `x^ndy/dx = y(log_e  y/x + log_e) ⇒ dy/dx = y/x^n(log^e*(y/x))` = f(x, y); let f(x, y) be a homogeneous function of degree 0, if n = 1.