Consider the following initial value problem, defined for t > 0: x' - 2x = t (t - w) e²w dw, 2w x(0) = = −1. Find the Laplace transform of the solution. X(s) = L {x(t)} = 1-83+22 1s 2s s²(5-2)² Obtain the solution x(t). x(t) = help (formulas) help (formulas)

Find the derivative of 2t-8 with respect to t. 4t+8 3t а. (4t+8)? 3t b. (4t+8)² 3 C. (t+2)? 3 d. (t+2)? 48 е. (t+2)?

Find the derivative of y with respect to x. y = cos -1 (5x²-4) A. - 10x OA√1-(543-4) О в. ○ C. O D. 1 + 10x (5x - 4)2 10x - (5x2 – 4)2 5 V1 + (5x - 4)2

Express the voltage waveform v(t) for figure 1 as a sum of unit step function for interval 0<t<7s and compute the derivative of ?(?) , and sketch its waveform.

1+Vx [8] Find y" of y =

Suppose solving an equation by Laplace transform results in Y(s) = s2 + 64' Evaluate y(T).

g(x) gʻ(5) = 2 x 13 1+t² dt

Find the derivative of the following: 1. y = = 5. y 3-2x 3+2x 2. y = (x²+4)²(2x³ — 1)³ - 3. If y = x² - 4x and x = √√2t² + 1, √2 find y'when t = V dy 4. Find given x = y√1-y² = 7. Find A -x 6. Find y" if x³ - 3xy + y³ = 1 dy dx dy given y = 8. Find x² + 2x 9. Find y", given y = given y = u³ +4, u = x x-1 10. Find y", given x² - xy + y² = 3 4√2 a. y' = ¹/2 (5-2√5) 5 b. c. y" dy dx -12 (3+2x)² 4-x 5 4(x-1)2 √1+u, u = √√x gy' = 2x(x² + 4) (2x³ − 1)² (13x³ + 36x - 2) e. y' dy d. = 6x²(x + 2)²(x + 1) dx f. y" h. j. dy dx i. y" = dy dx = = 8x-x3 (4-x²)2 -4xy (y²-x)³ = = B √1-y² 1-2y2 18 (x-2y)² 1 4√x+x√x