(dy/dt)−(dx/dt)=((t(dy/dt)+y)/(t(dx/dt)+x))


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Question Number 1030 by 123456 last updated on 20/May/15

$\frac{d y}{d t} - \frac{d x}{d t} = \frac{t \frac{d y}{d t} + y}{t \frac{d x}{d t} + x}$
	Answered by prakash jain last updated on 21/May/15

	$y = k_{1} t, x = k_{2} t$ $\frac{d y}{d t} = k_{1}, \frac{d x}{d t} = k_{2}$ $k_{1} - k_{2} = \frac{k_{1} t + k_{1} t}{k_{2} t + k_{2} t} = \frac{k_{1}}{k_{2}}$ $k_{1} - k_{2} = \frac{k_{1}}{k_{2}}$ $k_{2}^{2} - k_{1} k_{2} + k_{1} = 0$ $k_{2} = \frac{k_{1} \pm \sqrt{k_{1}^{2} - 4 k_{1}}}{2}$ $k_{1}^{2} - 4 k_{1} ⩾ 0 \Rightarrow k_{1} (k_{1} - 4) ⩾ 0 \Rightarrow k_{1} ⩽ 0 \lor k_{1} ⩾ 4$
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