Given x ̇ ̇ ̇(t) = u(t) with x(0) = a,x ̇(0) = b,x ̈(0) = c, express x(t)

[qumohammed]

Given x ̇ ̇ ̇(t) = u(t) with x(0) = a,x ̇(0) = b,x ̈(0) = c, express x(t)

Greetings All,

It has been more than 10 years since I did some math . So, I need your support. I'm trying to solve the following question:
Given x ̇ ̇ ̇(t) = u(t) with x(0) = a,x ̇(0) = b,x ̈(0) = c, express x(t) in terms of u and theseinitial conditions.



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I know that the integration of x ̇ ̇ ̇(t) = x ̈(t) +C1
Then integration of x ̈(t) = x ̇(t) +C2, integration of x ̇(t) = x(t)+C3.

Then what should I do next ?

 

[Ishuda]

It has been more than 10 years since I did some math . So, I need your support. I'm trying to solve the following question:
Given x ̇ ̇ ̇(t) = u(t) with x(0) = a,x ̇(0) = b,x ̈(0) = c, express x(t) in terms of u and theseinitial conditions.

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I know that the integration of x ̇ ̇ ̇(t) = x ̈(t) +C1
Then integration of x ̈(t) = x ̇(t) +C2, integration of x ̇(t) = x(t)+C3.

Then what should I do next ?

Well lets start with that "the integration of x ̇ ̇ ̇(t) = x ̈(t) +C1": This gives

\(\displaystyle \displaystyle \int_0^t\, x'''(s)\, ds\, =\,\)\(\displaystyle \displaystyle \int_0^t\, u(s)\, ds\,=\, x''(s)\, \bigg\rvert_0^t \, =\, x''(t)\, - \,x''(0)\,=\, x''(t)\, -\, a\)

Thus

\(\displaystyle x''(t)\, =\, \)\(\displaystyle \displaystyle \int_0^t\, u(s)\, ds\, +\, a\)

and C1 is -a.

Can you go from there?