A couple of beginner's differential equations

Hi,

I've just started on differential equations today. Could someone show me the ropes with the following problems?

1. Find the solution curves of the differential equation,

(x +1) dy/dx = x-1, through the origin, for x > -1

2. Water is leaking slowly out of a tank. The depth of the water after t hours is h metres, and these variables are related by a differential equation of the form dh/dt = -ae^-0.1t. Initially the depth of water is 6 metres, and after 2 hours it has fallen to 5 metres. At what depth will the level eventually settle down? Find an expression for dh/dt in terms of h.

Thanks folks!

Hi,

dy/dx = (x-1)/(x+1) = a/(x+1) + b

a + b + bx = x - 1 hence b = 1 and a =-2

dy/dx = -2/(x+1) + 1

hence y = -2.ln|x+1| + x + C

dh/dt = -ae^-0.1t

We need to work out h in terms of t (by integration)

h = 10.a.e^-0.1t + C

at t = 0 h= 6 so 10a + C = 6 (1)

at t = 2 h = 5 so 10a.e^-0.2 + C = 5 (2)

(1) - (2) <=> 10a.(1 - e^-0.2) = 1

hence a = 1/[10.(1-e^-0.2)]

which gives us C = 5 - 1/[e^0.2 - 1]

So you have h in terms of t, by working out the limit of h as t tends to + &#8734; you will find the height at which the water will settle itself. I found it to be C.

Oh I forgot dh/dt = -(h-C)/10

That's grand. It all makes sense now. Thank you.

By the way, C should be 6 - 1/[e^0.2 - 1], not 5 - 1/[e^0.2 - 1]

Cheers.