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 $\mathbf{D}\mathbf{=}\frac{\mathbf{d}}{\mathbf{dt}}$:

Given that,

$\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{=}\mathbf{4}\mathbf{x}\mathbf{+}\mathbf{y}$ …… (1)

$\frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}$ …… (2)

Let us rewrite this system of operators in operator form:

$\left(\mathbf{D}\mathbf{-}\mathbf{4}\right)\left[\mathbf{x}\right]\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}$ …… (3)

$\mathbf{2}\mathbf{x}\mathbf{+}\left(\mathbf{D}\mathbf{-}\mathbf{1}\right)\left[\mathbf{y}\right]\mathbf{=}\mathbf{0}$…… (4)

Multiply (D-1) on equation (3) and add with equation (4).

$$\begin{gathered} \left(\mathbf{D}\mathbf{-}\mathbf{1}\right)\left(\mathbf{D}\mathbf{-}\mathbf{4}\right)\left[\mathbf{x}\right]\mathbf{-}\left(\mathbf{D}\mathbf{-}\mathbf{1}\right)\mathbf{y}\mathbf{+}\left(\mathbf{2}\mathbf{x}\mathbf{+}\left(\mathbf{D}\mathbf{-}\mathbf{1}\right)\left[\mathbf{y}\right]\right)\mathbf{=}\mathbf{0} \\ \left({\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{D}\mathbf{+}\mathbf{4}\right)\left[\mathbf{x}\right]\mathbf{+}\mathbf{2}\mathbf{x}\mathbf{=}\mathbf{0} \\ \left({\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{D}\mathbf{+}\mathbf{6}\right)\left[\mathbf{x}\right]\mathbf{=}\mathbf{0} \end{gathered}$$

Since the corresponding auxiliary equation is ${\mathbf{r}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{r}\mathbf{+}\mathbf{6}\mathbf{=}\mathbf{0}$. The roots are r=2 and r=3.

Then, the general solution is $\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{2\mathbf{t}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}$ …… (5)

Substitute the equation (5) in equation (3).

$$\begin{gathered} \left(\mathbf{D}\mathbf{-}\mathbf{4}\right)\left[\mathbf{x}\right]\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0} \\ \mathbf{-}\mathbf{y}\mathbf{=}\mathbf{-}\left(\mathbf{D}\mathbf{-}\mathbf{4}\right)\left[\mathbf{x}\right] \\ \mathbf{y}\mathbf{=}\left(\mathbf{D}\mathbf{-}\mathbf{4}\right)\left[{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\right] \\ \mathbf{y}\mathbf{=}\frac{\mathbf{d}}{\mathbf{dt}}\left[{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\right]\mathbf{-}\mathbf{4}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}\mathbf{4}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}} \\ \mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{+}\mathbf{3}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\mathbf{-}\mathbf{4}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}\mathbf{4}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}} \\ \mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}} \\ \mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}                       …\left(6\right) \end{gathered}$$

Given:$\mathbf{x}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,}  \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}$

Now substitute the values in equations (5) and (6).

$$\begin{gathered} \mathbf{x}\left(\mathbf{t}\right)\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}} \\ \mathbf{x}\left(\mathbf{0}\right)\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\left(\mathbf{0}\right)}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\left(\mathbf{0}\right)} \end{gathered}$$

${\mathbf{c}}_{\mathbf{1}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}1$ …… (7)

$$\begin{gathered} \mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}} \\ \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{2}\left(\mathbf{0}\right)}\mathbf{-}{\mathbf{c}}_{\mathbf{2}}{\mathbf{e}}^{\mathbf{3}\left(\mathbf{0}\right)} \end{gathered}$$

$\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}\mathbf{-}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$ …… (8)

Solve the equations (7) and (8).

$$\begin{gathered} {\mathbf{c}}_{\mathbf{1}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{1}}\mathbf{-}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{0} \\ \mathbf{-}{\mathbf{c}}_{\mathbf{1}}\mathbf{=}\mathbf{1} \\ {\mathbf{c}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{1} \end{gathered}$$

Then,

$$\begin{gathered} {\mathbf{c}}_{\mathbf{1}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1} \\ \mathbf{-}\mathbf{1}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1} \\ {\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{2} \end{gathered}$$

Now substitute the values of c in equations (5) and (6).

$\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{+}\mathbf{2}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}$

$\mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{2}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}2{\mathbf{e}}^{\mathbf{3}\mathbf{t}}$

Thus, the required solution is:

$\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{+}\mathbf{2}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}$

$\mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{2}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}2{\mathbf{e}}^{\mathbf{3}\mathbf{t}}$