#### Shahirwan palat

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- Thread starter Shahirwan palat
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\(\displaystyle F(t)\equiv0\)

This gives us the homogeneous 2nd order linear IVP:

\(\displaystyle \frac{d^2x}{dt^2}+14\frac{dx}{dt}+k_1x=0\) where \(x(0)=-0.1,\,x'(0)=-4\)

Note that \(k_1=\dfrac{k}{M}\) and note the negative signs on the initial values, since we are presumably taking up to be in the positive direction. I am picturing a mass suspended under a spring, and the mass is pulled 10 cm (0.1 m) down from equilibrium in order to stretch, rather than compress, the spring.

If we wish to express the solution in trigonometric form, how should we write the characteristic roots?

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\(\displaystyle ar^2+br+c=0\)

Then, to express the solution in trigonometric form, we want to express the roots in complex form:

\(\displaystyle r=\frac{-b\pm i\sqrt{4ac-b^2}}{2a}\)

Do you see/understand why?

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Okay. I understand but may i ask on how to get the value for *k?*

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I don't think we have enough information to determine a numeric value for \(k\).Okay. I understand but may i ask on how to get the value fork?

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My Laplace transforms have gotten mighty rusty, as it's been 25 years. But, I would begin by stating:

\(\displaystyle \mathcal{L}\left\{\frac{d^2x}{dt^2}+14\frac{dx}{dt}+k_1x\right\}=\mathcal{L}\{0\}\)

Using the linearity of the operator, we have:

\(\displaystyle \mathcal{L}\{x''\}(s)+14\mathcal{L}\{x'\}(s)+k_1\mathcal{L}\{x\}(s)=0\)

Next, using the formulas for the derivatives and the initial conditions, we have:

\(\displaystyle \left[s^2Y(s)+0.1s+4\right]+14\left[sY(s)+0.1\right]+k_1Y(s)=0\)

Can you now solve for \(Y(s)\), and compute the inverse transform on the resulting rational function?

\(\displaystyle \mathcal{L}\left\{\frac{d^2x}{dt^2}+14\frac{dx}{dt}+k_1x\right\}=\mathcal{L}\{0\}\)

Using the linearity of the operator, we have:

\(\displaystyle \mathcal{L}\{x''\}(s)+14\mathcal{L}\{x'\}(s)+k_1\mathcal{L}\{x\}(s)=0\)

Next, using the formulas for the derivatives and the initial conditions, we have:

\(\displaystyle \left[s^2Y(s)+0.1s+4\right]+14\left[sY(s)+0.1\right]+k_1Y(s)=0\)

Can you now solve for \(Y(s)\), and compute the inverse transform on the resulting rational function?

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