Differential Equation = In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.

ODE = Ordinary Differential Equation

An ordinary differential equation is an equation which is defined for one or more functions of one independent variable and its derivatives. It is abbreviated as ODE. $y'=x+1$ is an example of ODE.

There are two ways to classify an ODE.

• Linear vs Non-linear

  • $f(x_1 + x_2) = f(x_1) + f(x_2)$
  • $f(kx) = kf(x)$
  • If a first order differential equation can be represented as $y' + p(x)y = r(x)$, it meets the two conditions above, it is linear.

• Homogeneous vs Non-homogeneous

  5-1)장 homogeneous와 non homogeneous의 차이 ( 제차와 비제차의 차이 )

  In $y' + p(x)y = r(x)$,

  • Homogeneous: $r(x) = 0$
  • Non-homogeneous: $r(x) \neq 0$

  왼쪽에 있는 $y' + p(x)y$ 은 시스템의 질량이나 물성치 등의 상태를 의미하고 $r(x)$는 외력을 의미하게 되며 $r(x) = 0$ 이라는 말은 외력이 없는 시스템의 고유한 상태방정식을 의미합니다.

• Steady state 와의 차이

Homogeneous ODE

We are going to solve $y' + p(x)y = 0$. We can get a general solution of $y$ from the following:

$$ \begin{split} y'+p(x)y = 0

& \Rightarrow \frac {dy} {dx} + p(x)y = 0

\\ & \Rightarrow \frac {dy} {y} = - p(x) dx

\\ & \Rightarrow \int \frac {dy} {y} = - \int p(x) dx

\\ & \Rightarrow \ln |y| = - \int p(x) dx + C

\\ & \Rightarrow y = e ^ {- \int p(x) dx + C}

\\ & \Rightarrow y = Ce ^ {- \int p(x) dx} ~~~ \cdots (0)

\end{split} $$

The solution of a first order linear homogeneous differential equation is  $y = Ce ^ {- \int p(x) dx}$.

Non-homogeneous ODE

5-3)장 : 1st linear non homogeneous ODE (1계 비제차 선형 상미분방정식 )

Let's solve $y' + p(x)y = r(x)$. But how? Let's use an integrator factor.

Let's multiply $F(x)$ to both sides.