Question

use the annihilator method to determinethe form of a particular solution for the given equation.$u\text{'}\text{'}-5u\text{'}+6u=cos2x+1$

Short answer

$u_p\left(x\right)=c_3+c_4\sin 2x+c_5\cos 2x$

Step-by-step solution

Solve the homogeneous of the given equation

The homogeneous of the given equation is

$$\begin{gathered} \left(D^2-5D+6\right)\left[u\right]=0 \\ \Rightarrow (D-2)(D-3)\left[u\right]=0 \end{gathered}$$

The solution of the homogeneous is

$u_h\left(x\right)=c_1{e}^{2x}+c_2{e}^{3x}$ (1)

Let$g\left(x\right)=\cos 2x$

Then

$$\begin{gathered} (D^2+2^2)\left[g\right]=0 \\ \Rightarrow (D^2+4)\left[g\right]=0 \end{gathered}$$

Let$h\left(x\right)=1$

Then

$D\left[h\right]=0$

Hence

$D(D^2+4)[g+h]=0$

Then, every solution to the given nonhomogeneous equation also satisfies

$D\left(D^2+4\right)(D-2)(D-3)\left[u\right]=0$

Then

$u\left(x\right)=c_1{e}^{2x}+c_2{e}^{3x}+c_3+c_4\sin 2x+c_5\cos 2x$(2)

is the general solution to this homogeneous equation

We know$u\left(x\right)=u_h+u_p$

Comparing (1) & (2)

$u_p\left(x\right)=c_3+c_4\sin 2x+c_5\cos 2x$