Question

Find the general solution to the differential equation 4d^2y/dx^2+4dy/dx+y=0

Answer 1

Charasteristic equation:

$4{k^2} + 4k + 1 = 0$

$D = 16 - 16 = 0$

$k = \frac{{ - 4}}{8} = - \frac{1}{2}$

Then

$y = {C_1}{e^{kx}} + x{C_2}{e^{kx}} = {C_1}{e^{ - \frac{1}{2}x}} + x{C_2}{e^{ - \frac{1}{2}x}}$

Answer: $y = {C_1}{e^{ - \frac{1}{2}x}} + x{C_2}{e^{ - \frac{1}{2}x}}$