Consider the differential equation
dy/dx = 2y4x
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)
1 Answer

dy/dx = 2y4x = 4x+2b 4x = 2b
y=2x+b ……dy/dx = 2
Therefore b = 1
d^2y/dx^2 = 2dy/dx 4 = 2(2y4x)4 = 4y – 8x – 4
d^2y/dx^2 < 0 , when x=0 & g(0) = 0
Thus (0,0) is a local maximum