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I'm trying to teach myself applied mathematics, specifically (simple) differential equations.

I've been solving them all fine, so far, but now I've come across second order DE's with a squared term, and I can't seem to get the right answer. I'll give an example:

d^2x/dt^2 = (dx/dt)^2

I've been using a substitution as follows:

u = dx/dt

Therefore:

du/dt = u^2

Then seperating:

du/u^2 = dt

Then integrating both sides:

-1/u = t + k

The inital conditions are if x=0, dx/dt (u) =1 and t=0

Therefore k = -1

Then:

-1/u = (t-1)

-1/(dx/dt) = (t-1)

-1/dx = (t-1)dt

Integrating again:

-x = (t^2)/2 - t + C

Initial conditions mean C = 0

Therefore, I get my answer to be:

x = t - (t^2)/2

The correct answer is: x = ln[1/(1-t)]

Could someone please show me where I'm going wrong? Is it just a simple algebraic mistake? Or is my method completely wrong? Thanks.

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# Second Order Seperable Differential Equations

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