2 Differential Equations questions

collegemathishard

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Oct 18, 2013
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1)Two skaters start on the x-axis, A at the origin
and B at the point (36,0). Suppose that A skates along the y-axis, that B
skates directly toward A at all times, and that B skates twice as fast as A:
How far will A travel before being caught by B?

2)
(xy√(x^2-y^2) + x)y′= y-x^2√(x^2-y^2)

I can't solve these so if you help me I will really be grateful. Thanks in advance!!
 
1)Two skaters start on the x-axis, A at the origin
and B at the point (36,0). Suppose that A skates along the y-axis, that B
skates directly toward A at all times, and that B skates twice as fast as A:
How far will A travel before being caught by B?

2)
(xy√(x^2-y^2) + x)y′= y-x^2√(x^2-y^2)

I can't solve these so if you help me I will really be grateful. Thanks in advance!!
Let v be A's speed in "x-units" per second. Since he starts at the origin and skates along the x axis, after t seconds, he will be at (vt, 0).

Let B's position at time t be (x, y). The vector from B to A, the vector from (x, y) to (vt, 0), is (vt- x)i- yj. That gives the direction in which B skates. A unit vector in that direction is \(\displaystyle \frac{vt- x}{\sqrt{(vt- x)^2+ y^2}}i+ \frac{y}{\sqrt{(vt- x)^2+ y^2}}j\) Since B skates twice as fast as A his speed is 2v so his velocity vector is \(\displaystyle 2v\left(\frac{vt- x}{\sqrt{(vt- x)^2+ y^2}}i+ \frac{y}{\sqrt{(vt- x)^2+ y^2}}j\right)\).

So we have \(\displaystyle \dfrac{dx}{dt}= \dfrac{2v(vt-x)}{\sqrt{(vt-x)^2+ y^2}}\) and \(\displaystyle \dfrac{dy}{dt}= \dfrac{2vy}{\sqrt{(vt-x)^2+ y^2}}\).
 
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Let v be A's speed in "x-units" per second. Since he starts at the origin and skates along the x axis, after t seconds, he will be at (vt, 0).

Let B's position at time t be (x, y). The vector from B to A, the vector from (x, y) to (vt, 0), is (vt- x)i- yj. That gives the direction in which B skates. A unit vector in that direction is \(\displaystyle \frac{vt- x}{\sqrt{(vt- x)^2+ y^2}}i+ \frac{y}{\sqrt{(vt- x)^2+ y^2}}j\) Since B skates twice as fast as A his speed is 2v so his velocity vector is \(\displaystyle 2v\left(\frac{vt- x}{\sqrt{(vt- x)^2+ y^2}}i+ \frac{y}{\sqrt{(vt- x)^2+ y^2}}j\right)\).

So we have \(\displaystyle \dfrac{dx}{dt}= \dfrac{2v(vt-x)}{\sqrt{(vt-x)^2+ y^2}}\) and \(\displaystyle \dfrac{dy}{dt}= \dfrac{2vy}{\sqrt{(vt-x)^2+ y^2}}\).

thanks but still I can't find how far a travels
 
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