The displacement x(t) of a cart of a mass-spring system is described by the diff

Question

The displacement x(t) of a cart of a mass-spring system is described by the differential equation d^2/dt^2 + 8 dx/dt + 15 x = 0 with the following Initial conditions: x (0) = 10. dx/dt (0) = 80. Calculate the maximum value of the displacement x(t) (for positive values of time t), round it off to three significant decimal digits, and provide the result. A student found that the result was as follows (your numerical answer must be written here) you must provide some intermediate results obtained by you while solving the problem above:

Explanation / Answer

The given equation is

x'' + 8x' + 15x = 0

The characteristic equation is r2 + 8r + 15 = 0

=> r2 + 5r + 3r + 15 = 0

=> r (r + 5) + 3 (r + 5) = 0

=> (r + 5) (r + 3) = 0

=> r = -5 and r = -3

Let x(t) = ae-5t + be-3t

When t = 0

x(0) = a + b = 10

Multiplying by 5

=> 5a + 5b = 50 (1)

Differentiating x(t)

x'(t) = -5ae-5t -3be-3t

When t = 0

x'(0) = -5a -3b = 80 (2)

Adding (1) and (2)

=> 2b = 130 => b = 65

Substituting in (1)

=> 5a + 5*65 = 50

=> 5a + 325 = 50

=> a = -275/5 = -55

Thus the solution is x(t) = -55e-5t + 65e-3t

Displacement x is maximum when x'(t) = 0

=> -5ae-5t -3be-3t = 0

Multiplying by e5t

=> -5a -3be2t = 0

=> e2t = -5a/3b

= -(5*-55) / (3*65)

= 275/195 = 1.41

=> 2t = ln 1.41 = 0.3435

=> t = 0.1718

Thus x(t) = -55e-5*0.1718 + 65e-3*0.1718

= -55e-0.859 + 65e-0.5154

= 85.533

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