Question

if ycos x = e^x, show that d^2y/dx^2-2tanxdy/dx-2y=0

Answer 1

just plug and chug.
y cosx = e^x
y = e^x secx
y' = e^x secx (1 + tanx) = y(1 + tanx)
y" = y' (1 + tanx) + y sec^2x

So, y" - 2tanx y' - 2y
= y' (1 + tanx) + y sec^2x - 2tanx (y(1+tanx)) - 2y
= y(1 + tanx)^2 + y (1+tan^2x) - 2y tanx - 2y tan^2x - 2y
= y + 2y tanx + y tan^2x + y + y tan^2x - 2y tanx - 2y tan^2x - 2y
= 0