Let y₁ = e³x and y2 = xe³x be solutions for a homogeneous DE. Then, is also a solution to the DE. Select one: None of these. y = 2xe-3x + 5e-3x O y = 2xe³x + 5e³x y = 2xe³x + 5e-3x O y = 2xe-3x + 5e³x The given set of functions: f₁(x) = 3x, f₂(x) = x-² and ƒ3(x) = x¹ is linearly independent on the interval (0,00). Select one: O True O False The function y = x ln (2x) is a particular solution for the nonhomogeneous differential equation xy" + xy - y = x + 1. Select one: O True O False The function y = 10x is a solution of the initial value problem y" - 100y = 0, y(0) = 1, y (0) = 10.

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A: Evaluating the definite integral

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A: We can find the answer as below.

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A: The objective of the question is slove the double integral and choose the correct option.

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A: Explanation of the answer is as follows

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A: The answer is given below:

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A: Given :r=3sinθr=cosθ

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A: To find the equation of tangent plane.

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