89 lines
3.2 KiB
Markdown
89 lines
3.2 KiB
Markdown
---
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date: 02.09.2024
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type: math
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---
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## Definitions and Basic Concepts
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### What is a Differential Equation?
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A **differential equation** is an equation that involves an unknown function and its derivatives. It describes how a quantity changes with respect to another (e.g., time, space). Differential equations are widely used in physics, engineering, economics, biology, and many other fields to model various phenomena.
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In mathematical terms, a differential equation can be written in the form:
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$$
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F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots\right) = 0,
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$$
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where $y = y(x)$ is the unknown function, and $\frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots$ are its derivatives.
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### What is an ODE?
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An **Ordinary Differential Equation (ODE)** is a type of differential equation that involves functions of only one independent variable and its derivatives. The general form of an ODE is:
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$$
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F\left(x, y, y', y'', \ldots, y^{(n)}\right) = 0,
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$$
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where $x$ is the independent variable, $y = y(x)$ is the dependent variable, and $y', y'', \ldots, y^{(n)}$ represent the first, second, and $n$-th derivatives of $y$ with respect to $x$.
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**Example:**
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A simple example of an ODE is the first-order linear ODE:
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$$
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\frac{dy}{dx} + p(x)y = q(x),
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$$
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where $p(x)$ and $q(x)$ are given functions.
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### What is a Linear and Homogeneous ODE?
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- A **linear ODE** is an ODE in which the dependent variable $y$ and its derivatives appear to the first power and are not multiplied together. A general $n$-th order linear ODE can be written as:
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$$
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a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y' + a_0(x)y = g(x),
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$$
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where $a_i(x)$ are functions of $x$ and $g(x)$ is a given function.
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- A **homogeneous ODE** is a special type of linear ODE where $g(x) = 0$. The general form is:
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$$
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a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y' + a_0(x)y = 0.
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$$
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**Example:**
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The second-order homogeneous linear ODE:
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$$
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y'' - 3y' + 2y = 0
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$$
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is homogeneous because the right-hand side is zero. It can be solved by finding the characteristic equation and determining the general solution.
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### What is a Particular Solution of ODEs?
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A **particular solution** of an ODE is a specific solution that satisfies both the differential equation and any given initial or boundary conditions. It is different from the **general solution**, which contains arbitrary constants that represent the family of all possible solutions to the differential equation.
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To find a particular solution, you substitute the initial or boundary conditions into the general solution and solve for the arbitrary constants.
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**Example:**
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Consider the ODE:
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$$
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y'' - 3y' + 2y = 0.
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$$
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The general solution is:
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$$
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y(x) = C_1 e^{2x} + C_2 e^x,
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$$
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where $C_1$ and $C_2$ are arbitrary constants. If we are given initial conditions $y(0) = 1$ and $y'(0) = 0$, we can substitute these into the general solution to find the values of $C_1$ and $C_2$, giving us a **particular solution**.
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**Steps to find a Particular Solution:**
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1. Find the general solution of the ODE.
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2. Use the given initial or boundary conditions to determine the values of the arbitrary constants in the general solution.
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3. Substitute these values back into the general solution to get the particular solution. |