50 / 40 : remove spaces within $$

50_ode/40_comparing_with_num_int.md

@@ -15,9 +15,9 @@ $$
 
 |  order  | Numerical Integration       | method | ODE Solver                    |
 |:---------:|:--------------------------------:|:--------:|:------------------------------------------------:|
-| 0th order | $$ F_k = f(x_k)\cdot \Delta x $$ |  Euler   | $$ x_{k+1} = x_{k} + \Delta t \cdot f(x_k, t_k) $$ |
-| 1st order | $$ F_k = \frac{\Delta x}{2}\left[f(x_k) + f(x_{k+1})\right] $$ |  Heun   | $$ x_{k+1} = x_{k} + \frac{\Delta t}{2} \left[f(x_k, t_k) + f(\hat{x}_{k+1}, t_{k+1})\right] $$ |
-| 2nd order | $$ F_k = \frac{\Delta x}{6}\left[f(x_k) + 4 \cdot f(x_{k+1}) + f(x_{k+2})\right] $$ |  Runge-Kutta   | $$ x_{k+1} = x_{k} + \frac{\Delta t}{6} \left[f(x_k, t_k) + 2 f(\hat{x}_{k+\frac{1}{2}}, t_{k+\frac{1}{2}})_1+ 2 f(\hat{x}_{k+\frac{1}{2}}, t_{k+\frac{1}{2}})_2 + f(\hat{x}_{k+1}, t_{k+1})\right] $$ |
+| 0th order | $$F_k = f(x_k)\cdot \Delta x$$ |  Euler   | $$x_{k+1} = x_{k} + \Delta t \cdot f(x_k, t_k)$$ |
+| 1st order | $$F_k = \frac{\Delta x}{2}\left[f(x_k) + f(x_{k+1})\right]$$ |  Heun   | $$x_{k+1} = x_{k} + \frac{\Delta t}{2} \left[f(x_k, t_k) + f(\hat{x}_{k+1}, t_{k+1})\right]$$ |
+| 2nd order | $$F_k = \frac{\Delta x}{6}\left[f(x_k) + 4 \cdot f(x_{k+1}) + f(x_{k+2})\right]$$ |  Runge-Kutta   | $$x_{k+1} = x_{k} + \frac{\Delta t}{6} \left[f(x_k, t_k) + 2 f(\hat{x}_{k+\frac{1}{2}}, t_{k+\frac{1}{2}})_1+ 2 f(\hat{x}_{k+\frac{1}{2}}, t_{k+\frac{1}{2}})_2 + f(\hat{x}_{k+1}, t_{k+1})\right]$$ |
 
 * Numerical integration and ODE solvers share a fundamental principle: approximating solutions using weighted averages. In numerical integration, these averages are of function values, while in ODE solvers, they are of slopes.<br>수치 적분과 상미분방정식 해법은 모두 기본적으로 가중 평균을 사용하여 해를 근사한다. 수치 적분에서는 함수 값의 평균을 사용하고, 상미분 방정식 해법에서는 기울기의 평균을 사용한다.
 * The accuracy of both numerical integration and ODE solvers is characterized by their order. Higher-order methods generally provide more accurate results but may be more computationally expensive.<br>수치적분과 상미분방정식 해법 모두 그 차수가 정확도를 결정한다. 차수가 높은 방법이 일반적으로 더 정확한 결과를 제공하지만, 계산 비용은 더 많이 들 수 있다.