dy-dt-dx-dt-t-dy-dt-y-t-dx-dt-x-

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