2nd order inhomogenous equations

[adichy]

Homework Statement

Find the general solution to the differential equation
$$\frac{d^2y}{dx^2}$$ -2*$$\frac{dy}{dx}$$ +2y =g(x)

where g(x) = −14 cos(2x) − 2 sin(2x)

Homework Equations

The Attempt at a Solution

ive found the homogenous solution which is
y=e^x (ae^ix +be^-ix)

what I am not sure of is for the particular intergral do i do
Csin(2x)+Dcos(2x)+Esin(2x)+Fcos(2x)
or is it just Csin(2x)+Dcos(2x)
please advise

thanks

edit:cant seem to get the latex to come out right basically its d^2y/dx^2 - 2(dy/dx) + 2y=0

 

[Mark44]

Tip: Use only one pair of tex tags, one at the beginning of the equation and one at the end.

Homework Statement

Find the general solution to the differential equation
$$\frac{d^2y}{dx^2}$$ -2*$$\frac{dy}{dx}$$ +2y =g(x)

where g(x) = −14 cos(2x) − 2 sin(2x)

Homework Equations

The Attempt at a Solution

ive found the homogenous solution which is
y=e^x (ae^ix +be^-ix)

You can also write this as y = e^x(A cos(x) + B sin(x))

what I am not sure of is for the particular intergral do i do
Csin(2x)+Dcos(2x)+Esin(2x)+Fcos(2x)
or is it just Csin(2x)+Dcos(2x)

For your particular solution, use y_p = Csin(2x)+Dcos(2x)

please advise

thanks

edit:cant seem to get the latex to come out right basically its d^2y/dx^2 - 2(dy/dx) + 2y=0

 

[hunt_mat]

I would start with a particular solution of:
$$P.I.=A\cos 2x+B\sin 2x$$
and the general solution is:
$$y=Ce^{\lambda_{1}x}+De^{\lambda_{2}x}$$
where the lambda are solutions of:
$$\lambda^{2}-2\lambda+2=0$$