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Differential Equations

I've got a few questions about some holiday homework I've got...

1. Show that x=Acos(wt+&#934 :wink:

, y=Bsin(wt+&#934 :wink:

is a solution to the system of differential equations: dy/dx=-y, dy/dx=4x, when w=2 (w>0) DONE
a) Thus show that A=1, B=2 and Φ=Pi/2 if x(o)=0 and y(0)=2 DONE
b) Hence, find the Cartesian equation of the ellipse satisfying the system
c) Draw a sketch of the ellipse

The equation of an ellipse is AP+BP=constant, but where do I get those values from? The A and B that I found earlier?

2. It is thought that the rate at which a rumour spreads in an office is jointly proportional to the number y, of people who have heard the rumour after t minutes, and the number N-y of those who have yet to hear it.
a) Explain why the differential equation dy/dx=ky(N-y) best describes this relationship- I don't know how to explain that...

3. Ley y= r cisΦ be any complex number written in polar form.
a) Show that y is a solution of the differential equation dy/dΦ=iy
I didn't know what that meant. However, I did get part b which was 'solve the differential equation....'

4. Suppose the system of differential equations x'(t)=-y, y'(t)=2x has a solution of the form x(t)=acos(wt+&#934 :wink:

and y(t)=Bsin(wt+&#934 :wink:

. DONE
a) Show that;
(1) w^2=2 DONE
(2) b^2=2a^2 DONE
(3) 2x^2 +y^2= b^2 Wasn't quitw sure how to do this one...

5. Verify that x(t)=2cost-3sint and y(t)=2sint +3cost is a solution for the system of differential equations x'=-y and y'=x
Show this solution describes a circle , centre (0,0) and radius root 13.

I got down to y(t)=Acost+Bsint and x(t)=Acost+Bsint. In the question the cos and sin have coefficients, yet we're not given initial conditions...So is that all I have to show? Also, how would you how that the solution describes a circle?

6. Verify that y^2= (1/4)x^2 -1 is a solution of the differential equation xy(dy/dx)=1+y^2
I have no idea what do do with this one...

Thankyou :smile:

Reply 1

meef cheese

Use the fact that (coswt)^2 + (sinwt)^2 = 1

Reply 2

galadriel100

e-unit

Use the fact that (coswt)^2 + (sinwt)^2 = 1

Ok...hmm...what do I do then? :redface:

Reply 3

steve10

galadriel100

Ok...hmm...what do I do then? :redface:

I imagine that suggestion was for question 1) b.
Think of it as: cos²(wt+&#934 :wink:

+ sin²(wt+&#934 :wink:

= 1 (look at your original parametric eqns)

2) Suppose you have company with N people, and after t mins, y people in the company have heard of the rumour. Then the rate at which the rumour is spreading can be given/described by y/t people per minute. However, since the rate is constantly changing then you would use the differential coefficient, dy/dx, to more accurately describe the rate of change .