Find the general solution of the equation and the specific solutions within the given interval sec(30) + 2 = 0, [0, 2x)

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A: identify the differential equation and solve it.

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A: x dydx+2 y =4x2dydx+2x y =4xComparing with Bernouli's equation;y'+P(x) y = Q(x)P(x)=2x, Q(x)=4x

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A: For orthogonal trajectory we replace dy/dy by -dx/dy

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A: I am attaching image so that you understand each and every step.

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A: I am attaching image so that you understand each and every step.

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A: Explanation of the answer is as follows

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A: g(x)=∫0xf(t)dtThenBy fundamental theorem of integral calculusg'(x)=ddx∫0xf(t)dt=f(x)g'(x)=2x2-1

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