make pdf file generation work

quarto/ODEs/differential_equations.qmd

@@ -72,13 +72,11 @@ $$
 The author's apply this model to  flu statistics from Hong Kong where:
 
 
-$$
 \begin{align*}
 S(0) &= 7,900,000\\
 I(0) &= 10\\
 R(0) &= 0\\
 \end{align*}
-$$
 
 In `Julia` we define these, `N` to model the total population, and `u0` to be the proportions.
 
@@ -133,13 +131,11 @@ The plot shows steady decay, as there is no mixing of infected with others.
 Adding in the interaction requires a bit more work. We now have what is known as a *system* of equations:
 
 
-$$
 \begin{align*}
 \frac{ds}{dt} &= -b   \cdot s(t)  \cdot  i(t)\\
 \frac{di}{dt} &= b  \cdot  s(t)  \cdot  i(t) - k  \cdot  i(t)\\
 \frac{dr}{dt} &= k  \cdot  i(t)\\
 \end{align*}
-$$
 
 Systems of equations can be solved in a similar manner as a single ordinary differential equation, though adjustments are made to accommodate the multiple functions.
 
@@ -282,12 +278,10 @@ We now solve numerically the problem of a trajectory with a drag force from air
 The general model is:
 
 
-$$
 \begin{align*}
 x''(t) &= - W(t,x(t), x'(t), y(t), y'(t)) \cdot x'(t)\\
 y''(t) &= -g - W(t,x(t), x'(t), y(t), y'(t)) \cdot y'(t)\\
 \end{align*}
-$$
 
 with initial conditions: $x(0) = y(0) = 0$ and $x'(0) = v_0 \cos(\theta), y'(0) = v_0 \sin(\theta)$.