- #1

- 44

- 0

d^2y/dx^2 +4dy/dx +8y = 4sin(2x) - 12cos(2x) y(0)=1 & y'(0)=-6

For the auxillary eqation I have (m + 2)^2 = sqrt -4

which gives m=-2-2i & m=-2+2i

Which gives y=e^-2x(Ccos(-2x) + Dsin(-2x))

Now to tackle particular integral.

Try y=acos(2x) + bsin(2x)

After differentiatiing twice and equating I have a=-5 and b=-9

Finally I get y=e^-2x(Ccos(-2x) + Dsin(-2x)) - 5cos(2x) -9sin(2x)

After aplying the initial conditions my final answer is

y=6e^-2xcos(-2x) - 18e^-2sin(-2x) - 5cos(2x) - 9sin(2x)

In all my working it is only e that is raised to the power of -2x, I know it looks like the whole statement is raised but it isnt, the e^-2x is multiplied by the trig expression.

James