I would like to know how do we solve d(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}x/dt^{2}= k' where k' is a constant i.e the task is to find x as a function of time ?

One way to approach this is to rewrite it as vdv/dx = k' where v=dx/dt and first find find v as a function of x and then rewrite v as dx/dt and then find x as a function of time .

I will present my attempt .

vdv/dx = k'

vdv= k'dx

∫vdv= ∫k'dx

v^{2}= 2k'x + 2C' where C' is a constant.

v=√(kx+C)

Now,v=dx/dt

dx/√(kx+C) =dt

∫dx/√(kx+C) =∫dt

x = (αt+β)^{2},where α and β are some constants.

Now ,I would like to know how do we solve the equation d^{2}x/dt^{2}= k' by direct integration.

∫(d^{2}x/dt^{2})dt = ∫k'dt

How to integrate the left hand side as it is a second order derivative ?

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# How to integrate second order derivative

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