I ONLY NEED NUMBER \" 5\" ANSWERED!!! x(0) = 4, 2. dx/dt =-x, y(0) = 2 3. dr/dt

Question

I ONLY NEED NUMBER " 5" ANSWERED!!!

x(0) = 4, 2. dx/dt =-x, y(0) = 2 3. dr/dt =-x, y@) = 0 4.dx/dt = 2y, dy/dt =2y: For each of the systems in Problems 1 through 6: (a) Find an equation of the form H(x,y) = c satisfied by the solutions of the given system. (b) Without using a computer, sketch some level curves of the function H(x, y). c) Fort> 0, sketch the trajectory corresponding to the given initial condition and indicate the direction of m otion for in- creasing t dy/dt = 2y: x(0) = 4, dy/a t = 8x; x(0) = 2, . 4, ,d-2), y(0) =-1 5. dx/dt = 2y, y(0) =-3 dy/dt = 8x; x(0) = 1, dy/dt=-2y; 1.ds/dt =-x, y(0) = 2 x(0) = 4,

Explanation / Answer

5.

x'=2y

Differentiating again w.r.t. t gives

x''=2y'=16x

x''=16x=4^2x

So general solution is

x=A exp(4t)+B exp(-4t)

y=x'/2=2 A exp(4t)-2B exp(-4t)

Using initial conditions gives

A+B=1

2A-2B=-3 ie A-B=-1.5

Solving gives

B=1.25,A=-0.25

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