19) For the ODE x(t) = e²t (c₁ cos(√3t) + c₂ sin(√3t) − sin (3t). cos (3t), let y = x' and 37 Y= (x²) (a) Couple the second order equation to write a first order system Y'= AY+b(t). let Y= 116 37

Q: 12 3. (sin x)y' +y cos r = r sin a,

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Q: Find f. a) f '''(x) = cos(x),    f(0) = 7,    f '(0) = 8,    f ''(0) = 8   b) A particle is moving…

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Q: If 7'(t) = (cos t, e2t, 3t²) and r(0) = (3, 1, –2), find the function 7 (t).

A: Given. f'(t) = (cos t , e2t , 3t² ) To calculate the vector function are we need to integrate the…

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A: Given,  f'''(x)=cosx, f(0)=9, f'(0)=8, f''(0)=6

Q: Find f. f"(x) = 5 + cos(x), f(0) = -1, f(9π/2) = 0

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Q: 3. Find the derivatives: a) f(x) = 3x³-4 2x²+3 b) DO NOT SIMPLIFY y = (x² + 6x + 2)(²+√x + 3) c)…

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Q: Let Find f(x) = y = f(x) = 5x + 18 cos x. d f(x) =

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Q: Let f(x) = cos(x). (a) On [0,2n) solve f(x) = 0. 4.

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Q: let y=[cos(2x-π)]3 Find y' at x=(π/6)

A: Compute the derivative as follows.

Q: F(s) = Le 4t cos(5t)}

A: We have to find  ℒe-4tcos5t Remember : ℒeatf(t)=Fs-a where  ℒf(t)=Fs

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Q: Are fi (x) = x, f2(x) = x², and f3(x) = 4x – 8x2 linearly independent or linearly dependent on (-00,…

A: Definition: A set of vectors are said to be linearly independent, if no vector can be represented as…

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Q: Find f. f"(8) = sin(0) + cos(8), F(0) = 5, f(0) = 4 F(8) -sin(0) – cos(0) + 5x + 6

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Q: // (cos (a? + y³)dA where D: {(x, y) | 4 < x² + y² < 49} Find

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Q: Find f if f"(x) = 2 + cos(x), f(0) = -10, f(x/2) = 4. %D %3D f(x) = iii

A: Recall: ∫xn dx= xn+1n+1+c; where c is an integration constant. ∫sinx dx= - cos(x)+c; where c is an…