【Advanced】Solving Ordinary Differential Equations with MATLAB

# Advanced篇: Solving Ordinary Differential Equations with MATLAB

## 1. Fundamentals of Ordinary Differential Equations (ODEs)

An Ordinary Differential Equation (ODE) is an equation that describes the relationship between an unknown function and its derivatives with respect to an independent variable. The form of a first-order ODE is y' = f(x, y), where y' denotes the derivative of y with respect to x. Higher-order ODEs are expressed as y^(n) = f(x, y, y', ..., y^(n-1)), with y^(n) denoting the nth-order derivative of y with respect to x.

## 2. Theories of Solving ODEs with MATLAB

### 2.1 Types and Solution Methods of ODEs

#### 2.1.1 First-order ODEs

The general form of a first-order ODE is:

y' = f(x, y)

Where:

* `y` is the unknown function, `x` is the independent variable.
* `f(x, y)` is a known function.

Methods for solving first-order ODEs include:

***Separation of Variables:** Separate the terms involving `x` and `y` on both sides of the equation and then integrate to obtain an implicit solution.
***Homogeneous Equation Method:** For a homogeneous equation `y' = g(y/x)`, let `v = y/x`, the equation transforms to `v' = h(v)`, and solving for `v` suffices.
***Integral Factor Method:** For a non-homogeneous equation `y' + p(x)y = q(x)`, let `μ(x) = e^(∫p(x)dx)`, the equation becomes `(μy)' = μq(x)`, and solving for `μy` suffices.

#### 2.1.2 Higher-order ODEs

The general form of a higher-order ODE is:

y^(n) + a_{n-1}y^(n-1) + ... + a_1y' + a_0y = f(x)

Where:

* `y` is the unknown function, `x` is the independent variable.
* `a_i` are constants.
* `f(x)` is a known function.

Solution methods for higher-order ODEs include:

***Characteristic Equation Method:** Solve the characteristic equation `r^n + a_{n-1}r^(n-1) + ... + a_1r + a_0 = 0`, construct solutions based on the nature of the roots.
***Variable Coefficient Method:** For non-homogeneous equations, first solve the homogeneous equation, then construct a particular solution, and finally add the homogeneous solution and the particular solution to obtain the general solution.
***Laplace Transform Method:** Apply the Laplace transform to both sides of the equation to obtain an algebraic equation, and then perform an inverse Laplace transform after solving the algebraic equation to obtain the solution.

### 2.2 Numerical Solution Methods

For ODEs that cannot be analytically solved, numerical methods can be employed.

**2.2.1 Explicit Methods**

Explicit methods use the value at the current time step to calculate the value at the next time step, such as:

***Euler's Method:**

y_{n+1} = y_n + h * f(x_n, y_n)

***Improved Euler's Method (Midpoint Method):**

y_{n+1} = y_n + h * f(x_n + h/2, y_n + h/2 * f(x_n, y_n))

**2.2.2 Implicit Methods**

Implicit methods treat the value at the next time step as the unknown in the equation, such as:

***Backward Euler's Method:**

y_{n+1} = y_n + h * f(x_{n+1}, y_{n+1})

***Implicit Midpoint Method:**

y_{n+1} = y_n + h * f(x_n + h/2, (y_n + y_{n+1})/2)

**2.2.3 Runge-Kutta Methods**

The Runge-Kutta method is a high-order explicit method, such as:

***RK4 Method (Classic Runge-Kutta Method):**

k_1 = h * f(x_n, y_n)
k_2 = h * f(x_n +