Calculus help!!??!?!?!?!?!?!?!?

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Consider the differential equation

dy/dx = 2y-4x

a) There is a value of b for which y=2x+b is a solution to the differential equation. Find this value of b. Justify your answer. (Hint: dy/dx is an equation for SLOPE and you can find the slope of the line in y=2x+b)

b) Let g be the function that satisfies the given differential equation with the initial condition g(0) = 0. It appears from the slope field that g has a local maximum at the point (0,0). Using the differential equation, prove analytically that this is so. (Hint: Use the second derivative test)

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1 Answer

  • dy/dx = 2y-4x = 4x+2b -4x = 2b

    y=2x+b ……dy/dx = 2

    Therefore b = 1

    d^2y/dx^2 = 2dy/dx -4 = 2(2y-4x)-4 = 4y – 8x – 4

    d^2y/dx^2 < 0 , when x=0 & g(0) = 0

    Thus (0,0) is a local maximum

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