3.46. Solve the system of differential equations:
a) $\left\{\begin{array}{l}\frac{d y_{1}(t)}{d t}+y_{1}(t)-y_{2}(t)=e^{t}, \\ \frac{d y_{2}(t)}{d t}+3 y_{1}(t)-2 y_{2}(t)=2 e^{t}, \\ y_{1}(0)=1, \quad y_{2}(0)=1, t>0 .\end{array}\right.$
b) $\left\{\begin{array}{l}\frac{d x}{d t}=x+2 y, \\ \frac{d y}{d t}=2 x+y+1 ;\end{array}\right.$ if $\left\{\begin{array}{l}x(0)=0, \\ y(0)=5 .\end{array}\right.$
Solution.

a) Transitioning to images, we have the system of linear algebraic equations:

$$
\left\{\begin{array}{l}
p \overline{y_{1}}(p)-1+\overline{y_{1}}(p)-\overline{y_{2}}(p)=\frac{1}{p-1} \\
\overline{y_{2}}(p)-1+3 \overline{y_{1}}(p)-2 \overline{y_{2}}(p)=\frac{2}{p-1}
\end{array}\right.
$$

or

$$
\left\{\begin{array}{l}
(p+1) \overline{y_{1}}(p)-\overline{y_{2}}(p)=\frac{p}{p-1} \\
3 \overline{y_{1}}(p)+(p-2) \overline{y_{2}}(p)=\frac{p+1}{p-1}
\end{array}\right.
$$

from which

$$
\overline{y_{1}}(p)=\overline{y_{2}}(p)=\frac{1}{p-1} \Rightarrow y_{1}(t)=y_{2}(t)=e^{t}
$$

b) Transitioning to images, we have the system of linear algebraic equations:

$$
\left\{\begin{array}{l}
p \bar{x}(p)=\bar{x}(p)+2 \bar{y}(p) \\
p \bar{y}(p)-5=2 \bar{x}(p)+\bar{y}(p)+\frac{1}{p}
\end{array}\right.
$$

or

$$
\left\{\begin{array}{l}
(1-p) \bar{x}(p)+2 \bar{y}(p)=0 \\
2 \bar{x}(p)+(1-p) \bar{y}(p)=-\frac{5 p+1}{p}
\end{array}\right.
$$

Solving this system for \(\bar{x}(p)\) and \(\bar{y}(p)\), we get:

$$
\bar{x}(p)=\frac{10 p+2}{p(p+1)(p-3)}, \bar{y}(p)=\frac{5 p^{2}-4 p-1}{p(p+1)(p-3)}
$$

To determine the originals \(x(t)\) and \(y(t)\), we use the decomposition theorem - formula 15 from table 3.1:

$$
\begin{aligned}
& \overline{f_{1}}(p)=10 p+2, \overline{f_{2}}(p)=p^{3}-2 p^{2}-3 p, \overline{f_{2}}(p)=3 p^{2}-4 p-3 ; \\
& p_{1}=0, p_{2}=-1, p_{3}=3 ; \\
& \frac{\overline{f_{1}}\left(p_{1}\right)}{\overline{f_{2}}\left(p_{1}\right)}=-\frac{2}{3}, \frac{\overline{f_{1}}\left(p_{2}\right)}{\overline{f_{2}^{\prime}}\left(p_{2}\right)}=-2, \frac{\overline{f_{1}}\left(p_{3}\right)}{\overline{f_{2}^{\prime}}\left(p_{3}\right)}=\frac{8}{3} .
\end{aligned}
$$

Thus, \(x(t)=-\frac{2}{3}-2 e^{-t}+\frac{8}{3} e^{3 t}\). Similarly, we find that \(y(t)=\frac{1}{3}+2 e^{-t}+\frac{8}{3} e^{3 t}\).

### 3.14.9. REFERENCE TABLES

Table 7 แщฺ 3.1

Basic Rules of Operational Calculus

| \os | \(f(t)\) | \(\bar{f}(p)=\int_{0}^{+\infty} e^{-pt} f(t) dt\) | Note |
| :---: | :---: | :---: | :---: |
| 1 | \(\frac{1}{2 \pi i} \int_{0}^{0 \cdots 1} e^{ppt} \bar{f}(p) dp\) | \(\bar{f}(p)\) | Inversion formula |
| 2 | \(\sum_{i=1}^{n} c_{i} f_{i}(t)\) | \(\sum_{i=1}^{n} c_{i} \overline{f_{i}}(p), \quad c_{i}=\) const | Linearity property |
| 3 | \(f'(t)\) | \(p \bar{f}(p)-f(0)\) | Image of the derivative |
| 4 | \(f^{(n)}(t), \quad n \geqslant 2\) | \(p^{n} \bar{f}(p)-\sum_{k=1}^{n} p^{n-k} f^{(n-k)}(0)\) |  |
| 5 | \(\int_{0}^{t} f(\tau) d \tau\) | \(\frac{1}{p} \bar{f}(p)\) | Image of the integral |
| 6 | \(-t f(t)\) | \(\bar{f}'(p)\) | Differentiation of the image |
| 7 | \((-1)^{n} t^{n} f(t), \quad n \geqslant 0\) | \(\bar{f}^{(n)}(p)\) |  |
| 8 | \(\frac{f(t)}{t}\) | \(\int_{p}^{\infty} \bar{f}(q) dq\) | Integration of the image |
| 9 | \(\int_{0}^{t} \frac{f(\tau)}{\tau} d \tau\) | \(\frac{1}{p} \int_{p}^{\infty} \bar{f}(q) dq\) | Integration of the original |
| 10 | \(\int_{t}^{+\infty} \frac{f(\tau)}{\tau} d \tau\) | \(\frac{1}{p} \int_{0}^{p} \bar{f}(q) dq\) |  |
| 11 | \(e^{kt} f(t)\) | \(\bar{f}(p-k)\) | Shift theorem |
| 12 | \(\left[\begin{array}{l}0, \quad t<\tau \\ f(t-\tau), \quad t \geq \tau\end{array}\right]\) | \(e^{-p \tau} \bar{f}(p)\) | Delay theorem |
| 13 | \(f(\lambda t), \quad \lambda>0\) | \(\frac{1}{\lambda} \bar{f}\left(\frac{p}{\lambda}\right)\) | Similarity theorem |
| 14 | \(\int_{0}^{t} f_{1}(\tau) f_{2}(t-\tau) d \tau\) | \(\overline{f_{1}}(p) \overline{f_{2}}(p)\) | Multiplication theorem of images |
| 15 | \(\sum_{k=1}^{+\infty} \frac{\overline{f_{1}}\left(p_{k}\right)}{\overline{f_{2}^{\prime}}\left(p_{k}\right)} e^{p_{k} t}\) | \(\frac{\overline{f_{1}}(p)}{\overline{f_{2}}(p)}=\frac{\overline{\bar{f}_{1}}(p)}{\prod_{k=1}^{+\infty} a_{k}\left(p-p_{k}\right)}\) | Decomposition theorem |

Table 3.2

Laplace Transforms of Some Functions

| ২o | \(f(t)\) | \(\bar{f}(p)\) |
| :---: | :---: | :---: |
| 1 | 1 | \(\frac{1}{p}, \quad \operatorname{Re} p>0\) |
| 2 | \(t^{\nu}\) | \(\frac{\Gamma(\nu+1)}{p^{\nu+1}}, \quad \nu>-1, \quad \operatorname{Re} p>0\) |
| 3 | \(t^{n}, n \in \mathbb{N}\) | \(\frac{n!}{p^{n+1}}, \quad \operatorname{Re} p>0\) |
| 4 | \(e^{\alpha t}\) | \(\frac{1}{p-\alpha}, \quad \operatorname{Re} p>\operatorname{Re} \alpha\) |
| 5 | \(\sin \omega t\) | \(\frac{\omega}{p^{2}+\omega^{2}}, \quad \operatorname{Re} p>\|\operatorname{Im} \omega\|\) |
| 6 | \(\cos \omega t\) | \(\frac{p}{p^{2}+\omega^{2}}, \quad \operatorname{Re