The solution of the differential equation ( 1 - x y - x 5 y 5 ) d x - x 2 ( x 4 y 4 + 1 ) d y = 0 given by (c is arbitrary constant)

Question

The solution of the differential equation $(1 - x y - x^{5} y^{5}) d x - x^{2} (x^{4} y^{4} + 1) d y = 0$ given by (c is arbitrary constant)

Smart Achievers · Accepted Answer

(1)

$The given equation is d x - x (y d x + x d y) = x^{5} y^{4} (y d x + x d y)$

$\Rightarrow \frac{d x}{x} = (1 + x^{4} y^{4}) d (x y)$ $\Rightarrow \ln x = x y + \frac{1}{5} x^{5} y^{5} + \ln c$

$\Rightarrow x = c e^{x y + \frac{1}{5} x^{5} y^{5}}$

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