Finding the maximum speed for a vector?

[mathgirl313]

So I have the equations x'(t) = 4*y(t) and y'(t)=-x(t). Going through all the steps, you get
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 (\alpha,\beta) then x(0) -[ \stackrel{2c1}{c2} ] = [ \stackrel{\alpha}{\beta} ]. I don't understand why this holds? (and that's my best attempt at a vector..sorry!)

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

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

Rearranging the equation we are constrained by we get: f(x,y) = g(x) = \sqrt{4*(\alpha)^2 + 16*(\beta)^2 - 3*x^2}, where x is in (-\sqrt{(\alpha)^2 + 4*(\beta)^2}, \sqrt{(\alpha)^2 + 4*(\beta)^2}. 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!

 

[HallsofIvy]

So I have the equations x'(t) = 4*y(t) and y'(t)=-x(t). Going through all the steps, you get
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 (\alpha,\beta) then x(0) -[ \stackrel{2c1}{c2} ] = [ \stackrel{\alpha}{\beta} ]. I don't understand why this holds? (and that's my best attempt at a vector..sorry!)

"Start" here is interpreted as "t= 0". Putting t= 0 in your equations for x and y, you get x(0)= 2c1 and y(0)= c2. Set those equal to \alpha and \beta.

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

Perhaps because it's not true. What is true is that since x(0)= \alpha and y(0)= \beta direct substitution gives x(0)^2/4+ y(0)^2= \alpha^2/4+ \beta^2. It is not true for general "t".

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

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

In order that that square root be a real number, the quantity in the root must not be negative. If x is not in that integral, it will be negative.

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!

 

[mathgirl313]

Thanks for the help! I pretty much get all of it now.

The last part though, I know the square root had to be greater than or equal to zero...but when I tried solving I got a different interval. I had 4/3 and 16/3 as coefficients for alpha and beta...but that was used for something different later in the problem...I'm not sure how this interval came up, I must have solved incorrectly or solved the wrong equation.