y\'\'[t]+y[t]=t , y[0]=1, y\'[0] = -1 The Laplace Transform of this problem came

Question

y''[t]+y[t]=t , y[0]=1, y'[0] = -1

The Laplace Transform of this problem came out to be Y[s] = [1/(s^2)+s-1]/[s^2+1]. I used mathematica to solve using the inverse laplace transform and the solution came out to be y[t] = t + cos[t] - 2sin[t].

I was looking at the solution found in the "Feedback Control of Dynamic Systems," 6th edition, by Franklin and Powell. It was problem (f) or 6, from Question 8 from Chapter 3. The solution from Cramster gave the solution to be y[t] = t + cos[t]. The problem with this is that they simplified Y[s] = [1/(s^2)+s-1]/[s^2+1] to
Y[s] = 1/[(s^2)*(s^2+1) + [s-1]/[s^2+1]. This does give the solution from Cramster, but the two Laplace Transforms are not equivalent, correct?

Explanation / Answer

If the efficiency is written as

then T

If the efficiency is written as

TH = W.net / Q.H = 1 – TLTH

then T

H is somewhere between 1500 K and 750 K and it is not a linear average.

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