Help solving this separation of variables?

Question

2(du/dt) = u^2 u( 0)= 1

Please give a step by step, I know that i am close to the right answer, i'm just missing a step thats throwing my answer off

i get to

t + k = (-2/u)

how do you plug in the u(0) though?

i also know that the answer is supposed to be

u = 1/(1-(1/2)t)

Answer 1

du/dt = u² / 2
u(t) = u³ / 6 + C
1 = 0 + C
C = 1
u(t) = u³ / 6 + 1

Answer 2

For separation of variables, you want to put everything of one variable, such as u, one side of the equal sign, and everything of the other variable, such as t, on the other side. Of course, the original expression needs to be capable of being separated in the first place. i.e. you can write the whole expression as a product like this: (something with only u)*(something with only t).

Anyway, ... putting each variable on either side, we get:

2du / dt = u^2 (given)
2du/(u^2) = dt (divide both sides by u^2, and multiply both sides by dt)

which simplifies to:
2u^(-2) du = dt

Integrating both sides leads to this:
2u^(-1) / (-1) + C2 = t + C1
-2 / u = t + C3 (C1 + C2 = sum of two constants is another constant, say C3)

Now solve for u in terms of t:
I get
u = -2 / (t + C3)

Now finally, to find C3, plug in your initial condition, u(0) = 1.
So when t = 0, u = 1.
1 = -2 / (0+C3)
1 = -2 / C3
C3 = -2

Plug in the value of C3 to get the final answer:
u = -2 / (t - 2)

Hope that helped... good luck! =)

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EDIT!!

So now you have:

t + k = (-2/u) . Great! You are almost there.
What does u(0) = 1 mean? u is a function of t, that is, using what you wrote above..
t + k = -2/u
tu + ku = -2
u(t+k) = -2
u = -2/(t+k)

So the variable u can be expressed in terms of t's and constants. Any function u in the form -2/(t+k) where k is a constant is a solution. However, there is only one solution that fulfills the initial value condition, u(0) = 1.

Your mission now is to determine the value of k such that when you plug in t=0 to the right hand side, you will get 1 on the left hand side. In other words...
u = -2/(t+k)
1 = -2/(0+k)
1 = -2/k
k = -2

Thus your final answer is u = -2/(t-2) = 2/[-(t-2)] =2/(2-t) = 1/(1-(1/2t)), which is what you said the answer was.

Answer 3

1. Seperate the variables accoring to their names. In other words, put 't' or 'u' on one side, and the other variable on the other side:

2 du = u^2 dt (multiply "dt" by both sides)
2/u^2 du = dt (divide u^2 by both sides)
integrate both sides

Plugin your original u(0) to solve for 'c' (the constant)!

Answer 4

this can be solved by integration
du/dt =u^2/2
dt/du =2/u^2
t=∫2/u^2 .du
t = -2 /u +c

as u(0)=1
0 =-2/1 +c
c = 2
t = -2/u +2

or
t-2 = -2/u
u =2/(2-t)
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about plugging
u(0)=1
if I say f(x) =x^2
& I say f(2) =4
that means I plug x with value=2 & got f(x)
here
u= 2/(2-t)
means
u(t) =2/(2-t)
so u(0)=1 means plug t =0 & u=1