Example 4.25. Find a particular solution of the equation

$$
y^{\prime \prime}+2 y^{\prime}-8 y=(12 x+20) e^{2 x}
$$

satisfying the initial conditions $y(0)=0, y^{\prime}(0)=1$.
Solution. 1. The characteristic equation $r^{2}+2 r-8=0$ has roots $r_{1}=-4, r_{2}=2$. Therefore,

$$
\bar{y}=C_{1} e^{-4 x}+C_{2} e^{2 x}
$$

2. The right-hand side of the given equation has the form (3): $n=1$, $P_{1}(x)=12 x+20, k=2$. Since $k=2$ is a single root of the characteristic equation, we seek the particular solution $y^{*}$ in the form

$$
y^{*}=x(A x+B) e^{2 x}=\left(A x^{2}+B x\right) e^{2 x}
$$

From this,

$$
\begin{gathered}
y^{*}=\left(2 A x^{2}+2 B x\right) e^{2 x}+(2 A x+B) e^{2 x}=\left(2 A x^{2}+2 A x+2 B x+B\right) e^{2 x} \\
y^{*}=\left(4 A x^{2}+4 A x+4 B x+2 B\right) e^{2 x}+(4 A x+2 A+2 B) e^{2 x}= \\
=\left(4 A x^{2}+8 A x+4 B x+2 A+4 B\right) e^{2 x}
\end{gathered}
$$

Substituting $y^{*}, y^{* \prime}$, and $y^{* \prime \prime}$ into the given equation, canceling $e^{2 x}$ from both sides, and combining like terms, we finally get

$$
12 A x+(2 A+6 B)=12 x+20
$$

Solving the system

$$
\begin{aligned}
12 A & =12, \\
2 A+6 B & =20,
\end{aligned}
$$

we find $A=1, B=3$. Therefore,

$$
y^{*}=\left(x^{2}+3 x\right) \mathrm{e}^{2 x} .
$$

Thus, the general solution of the given equation is

$$
y=\bar{y}+y^{*}=C_{1} e^{-4 x}+C_{2} e^{2 x}+\left(x^{2}+3 x\right) e^{2 x}
$$

3. To find the desired particular solution, we use the given initial conditions. We find the derivative of the general solution

$$
y^{\prime}=-4 C_{1} e^{-4 x}+2 C_{2} e^{2 x}+\left(2 x^{2}+8 x+3\right) e^{2 x} ;
$$

substituting the values $x=0, y=0, y^{\prime}=1$ into the expressions for the general solution and its derivative, we get the system of equations for finding $C_{1}$ and $C_{2}$ :

$$
\begin{aligned}
& 0=C_{1}+C_{2}, \\
& 1=-4 C_{1}+2 C_{2}+3 .
\end{aligned}
$$

From this, $C_{1}=\frac{1}{3}, C_{2}=-\frac{1}{3}$. Therefore, the desired particular solution is:

$$
y=\frac{1}{3} e^{-4 x}-\frac{1}{3} e^{2 x}+\left(x^{2}+3 x\right) e^{2 x}
$$

## Section 5

## Series

## 5.1. Basic Concepts

A numerical series is an expression of the form:

$$
a_{1}+a_{2}+a_{3}+\ldots+a_{n}+\ldots
$$

where the numbers $a_{1}, a_{2}, a_{3,} \ldots a_{n}$, called the terms of the series, form an infinite sequence.

A series is called convergent if the sequence of its partial sums

$$
\begin{aligned}
& S_{1}=a_{1}, \\
& S_{2}=a_{1}+a_{2}, \\
& S_{3}=a_{1}+a_{2}+a_{3}, \\
& S_{n}=a_{1}+a_{2}+a_{3}+\ldots+a_{n}
\end{aligned}
$$

has a finite limit as $n \rightarrow \infty$: $\lim _{n \rightarrow \infty} S_{n}=S$. This limit is called the sum of the convergent series. If the finite limit $\lim _{n \rightarrow \infty} S_{n}$ does not exist, the series is called divergent.