Simple second order question: Solution is clearly x(t) = t^5

[Tomski]

The problem d^2x/dt^2 = t^3 – 2t

Solution is clearly x(t) = t^5/20 –t^3/3+cx+d

However I’m slight confused about what the notation d^2x/dt^2 actually implies because to solve you would rearrange and then integrate both sides. Now I was under the impression that the integral of

d^2x would be xdx

hence you would get

xdx = t^4/4 – t^2 + c

which clearly isn’t the case because it produces an incorrect answer when you integrate again. Could some enlighten me on what I am doing incorrectly and possibly give me a layman's guide to the solution.

Thanks ever so much.

 

[Deleted member 4993]

d^2x/dt^2 is - by definition - the rate of change of dx/dt. Thus

\(\displaystyle \frac{d^2x}{dt^2} \, = \, \frac{d}{dt}(\frac{dx}{dt})\)

thus the first integration gives you:

\(\displaystyle \frac{dx}{dt}\)

and the second integration gives you

\(\displaystyle x(t)\)