📂Odinary Differential Equations

지수성장방정식/상수 계수를 갖는 1계 선형 동차 미분 방정식

Definition

In a first-order ordinary differential equation where the independent variable $t$ is not explicitly included in $f$, it is called an autonomous system or autonomous differential equation.

$$\dfrac{\mathrm{d}y}{\mathrm{d}t} = f(y)$$

Conversely, an equation in the following form is called a non-autonomous system.

$$\dfrac{\mathrm{d}y}{\mathrm{d}t} = f(y, t)$$

Explanation

The term autonomous system has a more dynamical sense, whereas autonomous differential equation feels more focused on the ordinary differential equation itself.

Since $y = y(t)$ holds, it is correct that $f$ contains information about $t$. However, when it impacts only by the value of $y = y(t)$, it is considered an autonomous system. It may be better understood as $y$ leading the system on its own (autonomously) without reliance on $t$.

One foundational and significant equation among autonomous systems is the following population model. It is also known as the exponential growth equation because the solution is an exponential function and it is used to model the phenomenon of population growth. By thinking about what function remains the same when differentiated once, you can understand why the exponential function is the answer.

Equation

$$\dfrac{dy}{dx} = \alpha y \tag{1}$$

The general solution of a first-order linear homogeneous differential equation with constant coefficients, as shown above, is as follows.

$$y=Ae^{\alpha x}$$

Here, $A$ is a constant.

Solution

Upon applying separation of variables to $(1)$, it is as follows.

$$\dfrac{dy}{dx} = \alpha y \implies \dfrac{1}{y} dy = \alpha dx$$

Integrating both sides, according to the derivative of the logarithm function, we have:

$$\ln y = a x + C$$

Here, $C$ is the constant of integration. Finally, taking the exponential function of both sides yields:

$$y=e^{\alpha x + C}=e^{\alpha x} e^{C}=Ae^{\alpha x}$$

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