Determine whether this is Homogenous or not (dy/dx)=(y/(y−2x)) .


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Question Number 181615 by Mastermind last updated on 27/Nov/22

$Determine whether this is Homogenous$ $or not$ $\frac{dy}{dx} = \frac{y}{y - 2 x}$ $.$
	Answered by hmr last updated on 27/Nov/22
	$(dy/dx) = ((((y/x)))/(((y/x))−2)) • z = ((y/x)) → (dy/dx) = z + x (dz/dx) z + x (dz/dx) = (z/(z − 2)) x (dz/dx) = (z/(z − 2)) − z x (dz/dx) = ((3z − z^2 )/(z − 2)) −(1/x) + ((z − 2)/(3z − z^2 )) (dz/dx) = 0 ∫−(1/x)dx + ∫((z − 2)/(3z − z^2 )) dz = c • ((z − 2)/(3z − z^2 )) = (a/(3−z)) + (b/z) z − 2 = az + b(3 − z) z − 2 = (a − b)z + 3b { ((a − b = 1)),((3b = −2 → b = ((−2)/3) → a = (1/3))) :} −ln∣x∣ + (1/3)ln∣3 − z∣ −(2/3)ln∣z∣= c −ln∣x∣ + (1/3)ln∣3 − (y/x)∣ −(2/3)ln∣(y/x)∣= c$
	$\frac{d y}{d x} = \frac{(\frac{y}{x})}{(\frac{y}{x}) - 2}$ $∙ z = (\frac{y}{x}) \to \frac{d y}{d x} = z + x \frac{d z}{d x}$ $z + x \frac{d z}{d x} = \frac{z}{z - 2}$ $x \frac{d z}{d x} = \frac{z}{z - 2} - z$ $x \frac{d z}{d x} = \frac{3 z - z^{2}}{z - 2}$ $- \frac{1}{x} + \frac{z - 2}{3 z - z^{2}} \frac{d z}{d x} = 0$ $\int - \frac{1}{x} d x + \int \frac{z - 2}{3 z - z^{2}} d z = c$ $∙ \frac{z - 2}{3 z - z^{2}} = \frac{a}{3 - z} + \frac{b}{z}$ $z - 2 = a z + b (3 - z)$ $z - 2 = (a - b) z + 3 b$ ${\begin{cases} a - b = 1 \\ 3 b = - 2 \to b = \frac{- 2}{3} \to a = \frac{1}{3} \end{cases}$ $- l n ∣ x ∣ + \frac{1}{3} l n ∣ 3 - z ∣ - \frac{2}{3} l n ∣ z ∣= c$ $- l n ∣ x ∣ + \frac{1}{3} l n ∣ 3 - \frac{y}{x} ∣ - \frac{2}{3} l n ∣ \frac{y}{x} ∣= c$
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