If y=3xe^2x, find dy/dx and d^2 y/dx^2 Showing that d^2 y/dx^2 -4 dy/dx +4y = 0?

Question

If y=3xe^2x, find dy/dx and d^2 y/dx^2 Showing that d^2 y/dx^2 -4 dy/dx +4y = 0?

Answer 1

If you mean

y=3xe^(2x)

Then using the product rule

y' = 3[ x*(2e^(2x) + (1)*e^(2x) ]

y' = 6xe^(2x) + 3e^(2x)

y'' = 6[ x* 2e^(2x) + (1)e^(2x) ] + 3*2e^(2x)

y'' = 12x * e^(2x) + 6e^(2x) + 6e^(2x)

y'' = 12x* e^(2x) + 12e^(2x)

y'' - 4y' + 4y = 0

(12x * e^(2x) + 12e^(2x) ) - 4(6xe^(2x) + 3e^(2x) ) + 4(3xe^2x)

24xe^(2x) - 24xe^(2x) + 12e^(2x)  - 12e^(2x) =0

0 = 0

Answer 2

dy/dx = 3e^2x + 6xe^2x

d^2y/dx^2 = 6e^2x + 6e^2x + 12xe^2x = 12e^2x + 12xe^2x

-4dy/dx = -12e^2x -24xe^2x

4y=12xe^2x

So d^2y/dx^2 -4dy/dx +4y = 0  ANSWER