Use the Laplace transform to solve the following initial-value problem for a first-order equation. y' - y = 6 cos(t), y(0) = 9

Q: Suppose h(x) = cos(tť² )dt Calculate h' (x).

A: Given: hx=∫0x3cosπt2dt.

Q: 10. Find the 87th derivative of y = cos(2x) at IT

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Q: Part 1: A. Differentiate the following functions with respect to x y = xe ²x (ii) y = 2x³ cos x…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: (4) 3. Use implicit differentiation to find dy/dx of the following: x² - 5xy + y³ = 21

A: As per policy only first question is answered. Please refer to the image below

Q: Prove that sinh−1 t = ln(t + √t2 + 1 ).

A:

Q: (d) Let y(x) = [tan-¹ (√)]4. Find y'.

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Q: Find F(s). F(s) = L{cos(8t) Ut - π)}

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Q: •T/4 t4 tant dt 2 + cos t -T/4

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Q: A yoyo bounces up and down in the vertical direction with its acceleration after tt seconds given by…

A: d^2y/dt^2 = a(t)

Q: Find the value(s) of ß for which y (t) = cos(ßt) satisfies y” + 100y = 0

A: Given that yt=cosβt satisfies the equation y''+100y=0. We have to find the values of β.

Q: dw where w = cos 8x sin 4y, x = dt and y =t". 4 Use the Chain Rule to find

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Q: The average rate of change of f(x) = cos(x) with respect to x as a changes from 0 to π is 2 OOO 2 -2…

A: We have to find the average rate of change.

Q: Find dx dx 2x r = f cos (?)dt 11

A: y = ∫x2πcos( t2 )dtwe want to find out dydx= ddx∫x2πcos( t2 )dt

Q: (a) Calculate the differential df of the function f = tan(x). Use it to find an = π/4, approximation…

A: Part (a):First, we need to calculate the differential df of the function f = tan(x). The derivative…