So I have the equations x'(t) = 4*y(t) and y'(t)=-x(t). Going through all the steps, you get(adsbygoogle = window.adsbygoogle || []).push({});

x(t) = c1*2*cos(2t) + c2*2*sin(2t) and

y(t) = -c1*sin(2t) + c2*cos(2t).

And of course, this can just can be written as the vector.

So after all that you are to find the maximum speed of the particle, which I have the whole solution to, I just don't understand parts of it.

If a particle starts at ([itex]\alpha[/itex],[itex]\beta[/itex]) then x(0) -[ [itex]\stackrel{2c1}{c2}[/itex] ] = [ [itex]\stackrel{\alpha}{\beta}[/itex] ]. I don't understand why this holds? (and that's my best attempt at a vector..sorry!)

And anyway c1 = [itex]\alpha[/itex] /2 and c2 = [itex]\beta[/itex] and using other earlier equations (x(t)^2)/4 + y(t)^2 = ([itex]\alpha[/itex] ^2)/4 + [itex]\beta[/itex] ^2. I don't understand

So we have the speed is given by [itex]\sqrt{16*y(t)^2 + x(t)^2}[/itex] = f(x,y), which we want to maximize, subject to x^2/4 + y^2 = ([itex]\alpha[/itex]^2) + ([itex]\beta[/itex])^2.

Rearranging the equation we are constrained by we get: f(x,y) = g(x) = [itex]\sqrt{4*(\alpha)^2 + 16*(\beta)^2 - 3*x^2}[/itex], where x is in (-[itex]\sqrt{(\alpha)^2 + 4*(\beta)^2}[/itex], [itex]\sqrt{(\alpha)^2 + 4*(\beta)^2}[/itex]. Why is x in this interval?

The rest of it from that point I understand. I just don't see why x is in that given interval, or why that original equation for x(0) holds? Thank you so much for any help!

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# Finding the maximum speed for a vector?

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