Example. Solve the Cauchy problem

$$
x^{\prime \prime}+4 x=\cos 2 t, \quad x(0)=1, \quad x^{\prime}(0)=-1
$$
Solution. Since $f(t)=\cos 2 t \eta(t)$ is the original, the solution $x(t)$ of the differential equation is also an original. Let's denote its image as $X(p)$.

## 1. Find the image of the left side of the equation.

By the theorem of differentiation of the original

$$
\begin{aligned}
x^{\prime}(t) & \longleftrightarrow p X(p)-x(0)=p X(p)-1 \\
x^{\prime \prime}(t) & \longleftrightarrow p^{2} X(p)-p x(0)-x^{\prime}(0)=p^{2} X(p)-p+1
\end{aligned}
$$

By the property of linearity

$$
x^{\prime \prime}+4 x \longleftrightarrow\left(p^{2}+4\right) X(p)-p+1
$$

2. Find the image of the right side of the equation

$$
\cos 2 t \longleftrightarrow \frac{p}{p^{2}+4}
$$

3. Form the operator equation

$$
\left(p^{2}+4\right) X(p)-p+1=\frac{p}{p^{2}+4}
$$

4. Solve the operator equation for $X(p)$:

$$
X(p)=\frac{p}{\left(p^{2}+4\right)^{2}}+\frac{p}{p^{2}+4}-\frac{1}{p^{2}+4}
$$

5. From the found image $X(p)$, restore the original $x(t)$:

$$
x(t)=\frac{1}{4} t \sin 2 t+\cos 2 t-\frac{1}{2} \sin 2 t
$$

6. By substituting $x(t)$ into the original differential equation, it becomes an identity. By calculating $x(0)$ and $x^{\prime}(0)$, we verify that $x(t)$ satisfies the initial conditions.

Answer. $\quad x(t)=\frac{1}{4} t \sin 2 t+\cos 2 t-\frac{1}{2} \sin 2 t$.

Conditions of the problem. Solve the Cauchy problems for differential equations.

1. $x^{\prime}+2 x=2-3 t$

$$
x(0)=0 \text {. }
$$

2. $x^{\prime}-x=t-1$,

$x(0)=0$.
3. $x^{\prime}+3 x=3 t-2$,

$x(0)=0$.
4. $x^{\prime \prime}-3 x^{\prime}+2 x=2 e^{3 t}$,

$x(0)=1, \quad x^{\prime}(0)=3$.
5. $x^{\prime \prime}+x^{\prime}=t \cos 2 t$,

$x(0)=0, \quad x^{\prime}(0)=0$.
6. $x^{\prime \prime}+9 x=1$,

$x(0)=0, \quad x^{\prime}(0)=0$.
7. $x^{\prime \prime}+4 x=\sin 3 t$

$x(0)=0, \quad x^{\prime}(0)=0$.
8. $x^{\prime \prime \prime}-x^{\prime \prime}=0$,

$x(0)=2, \quad x^{\prime}(0)=0, \quad x^{\prime \prime}(0)=1$.
9. $x^{\prime \prime \prime}+6 x^{\prime \prime}+11 x^{\prime}+6 x=0$,

$$
x(0)=1, \quad x^{\prime}(0)=-3, \quad x^{\prime \prime}(0)=9
$$

10. $x^{\prime \prime \prime}-3 x^{\prime \prime}+3 x^{\prime}-x=e^{t}, \quad x(0)=1, \quad x^{\prime}(0)=-1, \quad x^{\prime \prime}(0)=1$.

Answers. 1. $x(t)=1.75-1.5 t-1.75 e^{-2 t} . \quad 2 . \quad x(t)=-t$. 3. $x(t)=e^{-3 t}+t-1$. 4. $x(t)=e^{3 t}$. 5. $x(t)=-\frac{5}{9} \sin t+\frac{4}{9} \sin 2 t-\frac{t}{3} \cos 2 t$. 6. $x(t)=\frac{1}{9}(1-\cos 3 t)$. 7. $x(t)=\frac{3}{10} \sin 2 t-\frac{1}{5} \sin 3 t$. $8 . x(t)=1-t+e^{t}$. 9. $x(t)=e^{-3 t}$. 10. $x(t)=e^{t}\left(\frac{t^{3}}{6}+1-2 t+2 t^{2}\right)$.

### 2.13. Solution of systems of linear differential equations

Problem statement. Solve the Cauchy problem for systems of ordinary linear differential equations of the first order with constant coefficients:

$$
\dot{\mathbf{x}}=A \mathbf{x}(t)+\mathbf{f}(t)
$$

with initial conditions

$$
\mathbf{x}(0)=\mathbf{x}_{0}
$$

where $\mathbf{x}=\left\{x_{1}, \cdots, x_{n}\right\}$ is the vector of unknowns, $\mathbf{x}_{0}=\left\{x_{01}, \cdots, x_{n 0}\right\}$ is the vector of initial values,

$$
A=\left(\begin{array}{cccc}
a_{11} & a_{12} & \ldots & a_{1 n} \\
a_{21} & a_{22} & \ldots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \ldots & a_{n n}
\end{array}\right)
$$

is the matrix of coefficients and $\mathbf{f}(t)=\left\{f_{1}(t), \ldots, f_{n}(t)\right\}$ is the given vector function.

Plan of solution. If $f_{1}(t), \ldots, f_{n}(t)$ are originals, then the functions $x_{1}(t), \ldots, x_{n}(t)$ are also originals. Let's denote their images as $X_{1}(p), \ldots, X_{n}(p)$.

1. By the theorem of differentiation of the original and the property of linearity, find the images of the left and right sides of all equations of the system.
2. Form the system of operator equations

$$
p \mathbf{X}-\mathbf{x}_{0}=A \mathbf{X}(p)+\mathbf{F}(p)
$$

3. Solve the system of operator equations

$$
\mathbf{X}(p)=(p E-A)^{-1}\left(\mathbf{F}(p)+\mathbf{x}_{0}\right)
$$

where $E$ is the identity matrix of order $n$.

4. From the found images $\mathbf{X}(p)=\left\{X_{1}(p), X_{2}(p), \ldots, X_{n}(p)\right\}$, restore the originals $x_{1}(t), x_{2}(t), \ldots, x_{n}(t)$.
5. Verify that $x_{1}(t), x_{2}(t), \ldots, x_{n}(t)$ satisfy the original system of differential equations and the initial conditions.

Write the answer.