Need Verification: [x(t)]^2 - 2[x'(t)]^2 = c where c is some

Hi everyone. I've done a differential equation problem but am unsure if I'm right. I would appreciate it if someone could look over what I did ...

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Find x(t) for:

. . .[x(t)]^2 - 2[x'(t)]^2 = c

...where c is some constant.

Differentiate both sides with respect to t:

. . .2x'(t) - 4x"(t) = 0

. . .1/2 x'(t) = x"(t)

. . .x"(t) - 1/2 x'(t) = 0

Let u'(t) = x"(t) and u(t) = x'(t). Then:

. . .u'(t) - 1/2 u(t) = 0

. . .d [ e^[-t/2) u(t) ] / dt = 0

. . .e^[(-t/2)]t u(t) = c

...where c is a constant

. . .u(t) = x'(t) = c e^[(t/2)]

Integrate both sides to get:

. . .x(t) = 2c e^[(t/2)] + k

...where k is a constant of integration

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There. Hope it's right. It looks right though.

Re: Need Verification: [x(t)]^2 - 2[x'(t)]^2 = c where c is

uberathlete said:

Differentiate both sides with respect to t:

2x'(t) - 4x"(t) = 0

Click to expand...

You seem to have forgotten the Chain Rule.

chain rule?

[x(t)]² - 2[x'(t)]² = C

2[x(t)]x'(t) - 4[x'(t)]x"(t) = 0

2x'(t)[x(t) - 2x"(t)] = 0

x'(t) = 0 or x(t) - 2x"(t) = 0

x(t) is a constant or x(t) = ke^t/sqrt(2)

re

Argh! Chain Rule!!! Hmm .. ok, so there could be two solutions. Thank you tkhunny and skeeter.