z" +7.84x = 5 cos(3t), (0) = r'(0) = 0

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A: We need to make it proper to solve this integration.

A: We can find the value of f(x) as below.

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A: Differentiation chain rule

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A: By applying the product rule for the differentiation.

A: Objective is to find f.

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A: Given, ∫-π3π232cosx+1dx We have to evaluate the definite integral.

A: Given :y''-y'-y=cos5t-sin5ty(0)=2 and y'(0)=5

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A: Given that, 4y''-4=cos-1x Now let's take, 4y''-4=0 Take m=ddx, 4m2-4y=04m2-4=0m=±1 So the homogenous…

A: Given, f"(θ)=sinθ+cosθ f(0)=5 and f'(0)=2

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A: We have to find

A: Compute the derivative as follows.

A: Given problem:- f''(θ) = sin(θ) + cos(θ) , f(0) =2 , f'(0) =2  To find:- f(θ)

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A: Differential equation: f'''(x)=cosx f(0)=6, f'(0)=3 f''(0)=5

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A: This equation solve by separation variable

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A: Solution :-

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A: Just use the basic integration of trigonometric and then find the value of integration constant from…