2. (Laplace Transforms) (a) Use the definition of the Laplace transform to compute L{cos(a + t)}, where a € R. What's the domain of L{cos(a + t)}? (b) Use Laplace transforms to solve y" - a²y = cos(a + t) where y(0) = 1 and y' (0) = 2. (c) Is the solution in part (b) unique? Justify using a theorem.

Q: 3. g(t) = arcsin (E - find g'(t) A. (4-t+) В. (4-t+)ž C. - 2t5 (4-t4)ž 2t D. (4-t+)5

A: 3-Given- gt=arcsint22-t24-t4 To Find- The value of g't.

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Q: (1) (II) 10 F. 10 +t 10 10 -10 -10 (II) (IV) 10 10 斗t 10 t 10 -10 -10 a) y = 10e-t/2 cos 2nt b) y =…

A:

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Q: 1. (a) Construct the first Taylor approximant to f: (−1,1) → R, ƒ(x) = sin¯¹ x about a = 11. (b) Use…

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Q: (1) If y = sin(tan-x), prove that %3D (1+ x²)?y" + 2x(1 + x²)y' +y = 0.

A: Since you have asked multiple questions in single request.so we will answering only first question.

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A: Using derivative formula.

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Q: Q3) for f(t) = sec?(6t – 1) and g(t) = tan?(6t - 1) show that %3D f'(t) = g'(t).

A:

Q: Find the derivative with respect to x. (c) y = v7x – x4 (d) y = (1+ sin 7.x)-5 (a) y = (5x – 2)*…

A: This question is related to derivative, we will solve it using given information.

Q: 5. Find L{f(t)} a. f(t) = sin?(t) + t b. f(t) = n² – 8e¬2t + (et + 1)² | c. f(t) = F, sin yt -cos…

A: As per our limit we can solve only first three subdivisions if you want the answer for other…

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Q: 3. Let f(x)= :/ arcsin(t) dt and g(x)= / arccos(t) dt. 0. 0. I f'(x) g' (x) dx =? 4 - T (a) 2 - T…

A: Given Applying derivative with respect to x: Similarly: