If y = e^x (sin x + cos x), then show that d^2y/dx^2-2(dy/dx)+2y=0 ​

Question

If y = e^x (sin x + cos x), then show that \(\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\)

Answer 1

Given,

y = e^x (sin x + cos x)

Answer 2

Since y = e^x (sinx + cosx)

dy/dx = e^x (sinx + cosx) +e^x (cosx - sinx)

dy/dx = y + e^x (cosx - sinx)

d²y/ dx² = dy/dx + e^x (cosx - sinx) +e^x(-sinx - cosx)

d²y/dx² = dy/dx + dy/dx - y - y

d²y/dx² = 2dy/dx -2y

d²y/dx² - 2dy/dx + 2y = 0

Hence proved