91Ó°ÊÓ

Elimination Procedure for 2 × 2 Systems:

To find a general solution for the system

$$\begin{gathered} {\mathbf{L}}_{\mathbf{1}}\left[\mathbf{x}\right]\mathbf{+}{\mathbf{L}}_{\mathbf{2}}\left[\mathbf{y}\right]\mathbf{=}{\mathbf{f}}_{\mathbf{1}}\mathbf{,} \\ {\mathbf{L}}_{\mathbf{3}}\left[\mathbf{x}\right]\mathbf{+}{\mathbf{L}}_{\mathbf{4}}\left[\mathbf{y}\right]\mathbf{=}{\mathbf{f}}_{\mathbf{2}}\mathbf{,} \end{gathered}$$

Where${\mathbf{L}}_{\mathbf{1}}\mathbf{,}{\mathbf{L}}_{\mathbf{2}}\mathbf{,}{\mathbf{L}}_{\mathbf{3}}\mathbf{,}$ and${\mathbf{L}}_{\mathbf{4}}$ are polynomials in $D=\frac{d}{dt}$:

1. Make sure that the system is written in operator form.
2. Eliminate one of the variables, say, y, and solve the resulting equation for x(t). If the system is degenerating, stop! A separate analysis is required to determine whether or not there are solutions.
3. (Shortcut) If possible, use the system to derive an equation that involves y(t) but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for x(t) into this equation to get a formula for y(t). The expressions for x(t), y(t) give the desired general solution.
4. Eliminate x from the system and solve for y(t). [Solving for y(t) gives more constants----in fact, twice as many as needed.]
5. Remove the extra constants by substituting the expressions for x(t) and y(t) into one or both of the equations in the system. Write the expressions for x(t) and y(t) in terms of the remaining constants.

Given that,

$$\begin{gathered} x\text{'}=4x-4z                  ......\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ y\text{'}=4y-2z                  ......\mathbf{\left(}\mathbf{2}\mathbf{\right)} \\ z\text{'}=-2x-4y+4z      ......\mathbf{\left(}\mathbf{3}\mathbf{\right)} \end{gathered}$$

Let us rewrite the given system of equations into operator form.

$$\begin{gathered} \left(D-4\right)\left[x\right]+4\left[z\right]=0      ......\mathbf{\left(}\mathbf{4}\mathbf{\right)} \\ \left(D-4\right)\left[y\right]+2\left[z\right]=0      ......\mathbf{\left(}\mathbf{5}\mathbf{\right)} \\ 2\left[x\right]+4\left[y\right]+\left(D-4\right)\left[z\right]=0      ......\mathbf{\left(}\mathbf{6}\mathbf{\right)} \end{gathered}$$

Multiply D-4 on equation (6). Then, substitute equation (4) and (5).

$$\begin{gathered} 2\left(D-4\right)\left[x\right]+4\left(D-4\right)\left[y\right]+{\left(D-4\right)}^2\left[z\right]=0 \\ -8\left[z\right]-8\left[z\right]+{\left(D-4\right)}^2\left[z\right]=0 \\ {\left(D-4\right)}^2\left[z\right]-16\left[z\right]=0 \\ \left(D^2-8D+16-16\right)\left[z\right]=0 \end{gathered}$$

$$\begin{gathered} \left(D^2-8D\right)\left[z\right]=0 \\ \left(D^2-8D\right)\left[z\right]=0      ......\mathbf{\left(}\mathbf{7}\mathbf{\right)} \end{gathered}$$

Since, the auxiliary equation to the corresponding homogeneous equation is:

$r^2-8r=0$

. The roots are r =0 and r = 8.

Then, the general solution of z is

$z\left(t\right)=c_1+c_2{e}^{8t}      ......\mathbf{\left(}\mathbf{8}\mathbf{\right)}$

Now substitute equation (8) in equation (5).

$$\begin{gathered} \left(D-4\right)\left[y\right]+2\left[z\right]=0 \\ \left(D-4\right)\left[y\right]=-2\left[z\right] \\ \left(D-4\right)\left[y\right]=-2\left[c_1+c_2{e}^{8t}\right] \\ \left(D-4\right)\left[y\right]=-2c_1-2c_2{e}^{8t} \end{gathered}$$

$\left(D-4\right)\left[y\right]=-2c_1-2c_2{e}^{8t}      ......\mathbf{\left(}\mathbf{9}\mathbf{\right)}$

So, the complementary solution of the differential equation is$y_h\left(t\right)=c_3{e}^{4t}      ......\mathbf{\left(}\mathbf{10}\mathbf{\right)}$

Let us assume that function: $y_p\left(t\right)=A+B{e}^{8t}      ......\mathbf{\left(}\mathbf{11}\mathbf{\right)}$

Find the derivation of equation (11).

$D\left[y_p\left(t\right)\right]=8B{e}^{8t}$

Use the derivation in equations (9) to get,

$$\begin{gathered} \left(D-4\right)\left[y\right]=-2c_1-2c_2{e}^{8t} \\ \left(D-4\right)\left[A+B{e}^{8t}\right]=-2c_1-2c_2{e}^{8t} \\ 8B{e}^{8t}-4A-4B{e}^{8t}=-2c_1-2c_2{e}^{8t} \\ -4A+4B{e}^{8t}=-2c_1-2c_2{e}^{8t} \end{gathered}$$

Now, equalise the like terms.

$$\begin{gathered} 4A=2c_1 \\ A=\frac{c_1}{2} \end{gathered}$$

Then,

$$\begin{gathered} 4B=-2c_2 \\ B=-\frac{c_2}{2} \end{gathered}$$

So, $y_p\left(t\right)=\frac{c_1}{2}-\frac{c_2}{2}{e}^{8t}$.

Then,$y\left(t\right)=\frac{c_1}{2}-\frac{c_2}{2}{e}^{8t}+c_3{e}^{4t}      ......\mathbf{\left(}\mathbf{12}\mathbf{\right)}$

Now substitute the equation (8) and (12) in equation (6)

$$\begin{gathered} 2\left[x\right]+4\left[y\right]+\left(D-4\right)\left[z\right]=0 \\ 2\left[x\right]=-4\left[y\right]-\left(D-4\right)\left[z\right] \\ x=-2\left[y\right]-\frac{1}{2}D\left[z\right]+2\left[z\right] \\ =-2\left[\frac{c_1}{2}-\frac{c_2}{2}{e}^{8t}+c_3{e}^{4t}\right]-\frac{1}{2}D\left[c_1+c_2{e}^{8t}\right]+2\left[c_1+c_2{e}^{8t}\right] \end{gathered}$$

$$\begin{gathered} =-2\frac{c_1}{2}+2\frac{c_2}{2}{e}^{8t}-2c_3{e}^{4t}-\frac{8}{2}{c}_2{e}^{8t}+2c_1+2c_2{e}^{8t} \\ =c_1-c_2{e}^{8t}-2c_3{e}^{4t} \end{gathered}$$

So, the solution is founded.