Show that if ycosx = e^x then d^2y/dx^2 - 2 tanx dy/dx - 2y=0

Answer 1

Given y * cos x = e^x

By implicit differentiation, the first derivative is

-y * sin x + y' * cos x = e^x

By implicit differentiation, the second derivative is

-y cos x - y' sin x - y' sin x + y'' cos x = e^x

-y cos x - 2 * y' sin x + y'' * cos x = e^x <--- combines the two like terms in the middle on left side

-y cos x - 2 * y' sin x + y'' * cos x = y * cos x <--- substitutes the original function y*cosx = e^x

-2y cos x - 2 * y' sin x + y'' * cos x = 0 <--- subtracts y*cos x from both sides

-2y - 2 * y' tan x + y'' = 0 <--- divides both sides by cos x

y'' - 2*tanx * y' - 2y = 0

end of proof