Hey would you use the Method of Undetermined Coefficients (Method of Good Guessing) to find a particular sol'n to this DE
$\displaystyle y''-5y'+6y=7e^{2x}+cos(2x)$
And you would have to split the particular solution up because the non homogeneous part is the sum of two terms?
In other words find Yp for $\displaystyle y''-5y'+6=7e^{2x}$ then find Yp for $\displaystyle y''-5y'+6=cos(2x)$ and the two of them together make up the particular solution for the original DE?
Solve the homogeneous DE:
$\displaystyle y'' - 5y' + 6y = 0$
Characteristic Equation:
$\displaystyle m^2 - 5m + 6 = 0$
$\displaystyle (m - 2)(m - 3) = 0$
$\displaystyle m = 2$ or $\displaystyle m = 3$.
So $\displaystyle y_c = Ae^{2x} + Be^{3x}$.
Now, notice that we already have a term of the form $\displaystyle e^{2x}$ in $\displaystyle y_c$. So all we need to worry about now in the particular solution is $\displaystyle C_1\cos{2x} + C_2\sin{2x}$.
$\displaystyle y_p = C_1\cos{2x} + C_2\sin{2x}$
$\displaystyle y_p' = 2C_2\cos{2x} - 2C_1\sin{2x}$
$\displaystyle y_p'' = -4C_2\sin{2x} - 4C_1\cos{2x}$.
Substitute these into your DE and solve for $\displaystyle C_1$ and $\displaystyle C_2$.
Then $\displaystyle y = y_c + y_p$.
From the homogenous solution to the differential you find that solutions are:
$\displaystyle y=c_1e^{2x}+c_2e^{3x}$.
However, when you go to solve for the two particular solutions to the differential, you notice that you already have an exponential to the 2x as a solution of your homogenous. So you need to multiple your original "guess" by $\displaystyle x^{k}$, where k is the smallest exponent needed to get a unique guess that is not a solution to our homogenous differential.
Therefore I would solve my particular as:
$\displaystyle y_p = Axe^{2x}$
$\displaystyle y'_p = Ae^{2x}+2Axe^{2x} \Rightarrow Ae^{2x}[1+2x]$
$\displaystyle y''_p=2Ae^{2x}[1+2x]+Ae^{2x}[2] \Rightarrow 4Axe^{2x}+4Ae^{2x}$
Now substituting into your original d.e.:
$\displaystyle 4Axe^{2x}+4Ae^{2x}-5(Ae^{2x}+2Axe^{2x})+6(Axe^{2x})=7e^{2x} \Rightarrow$
$\displaystyle 4Axe^{2x}+4Ae^{2x}-5Ae^{2x}-10Axe^{2x}+6Axe^{2x}=7e^{2x}$
Terms cancel out leaving:
$\displaystyle -Ae^{2x}=7e^{2x}$
Therefore our particular solution for our exponential is $\displaystyle -7xe^{2x}$