Find the general solution of y\'\' - y\' - 2y = 6x + ex Step by step please Loca

Question

Find the general solution of y'' - y' - 2y = 6x + ex Step by step please Locate all relative maxima, minima and saddle points for f(x, y) = x3 + y2 - 12x + 6y - 7. Step by step please

Explanation / Answer

y = e^rx =>y' = re^rx and y'' = r^2e^rx Homogeneous equation y''-y'-2y = 0 r^2e^rx-re^rx-2e^rx = 0 =>e^rx(r^2-r-2) = 0 =>r^2-r-2 = 0 =>r^2-2r+r-2 = 0 =>r = 2, -1 Therefore y = Ae^-x + Be^2x + c f(x,y) = x^3+y^2-12x+6y-7 = 0 =>fx = 3x^2-12 = 0 => x = 2, -2 fy = 0 => 2y+6 = 0 => y = -3 fxx = 6x =>fxx > 0 if x = 2.......minimum fxx

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