use quarto, not Pluto to render pages

CwJ/ODEs/odes.jmd

@@ -93,8 +93,11 @@ x'(t) = v(t) = v_0 + a (t - t_0), \quad x(t_0) = x_0.
 Again, we can integrate to get an answer for any value $t$:
 
 ```math
-x(t) - x(t_0) = \int_{t_0}^t \frac{dv}{dt} dt = (v_0t + \frac{1}{2}a t^2 - at_0 t) |_{t_0}^t =
-(v_0 - at_0)(t - t_0) + \frac{1}{2} a (t^2 - t_0^2).
+\begin{align*}
+x(t) - x(t_0) &= \int_{t_0}^t \frac{dv}{dt} dt     \\
+&= (v_0t + \frac{1}{2}a t^2 - at_0 t) |_{t_0}^t    \\
+&= (v_0 - at_0)(t - t_0) + \frac{1}{2} a (t^2 - t_0^2).
+\end{align*}
 ```
 
 There are three constants: the initial value for the independent variable, $t_0$, and the two initial values for the velocity and position, $v_0, x_0$.  Assuming $t_0 = 0$, we can simplify the above to get a formula familiar from introductory physics:
@@ -174,13 +177,15 @@ A graph of the solution for $T_0=200$ and $T_a=72$ and $r=1/2$ is made
 as follows. We've added a few line segments from the defining formula,
 and see that they are indeed tangent to the solution found for the differential equation.
 
-```julia; hold=true; echo=false
-T0, Ta, r = 200, 72, 1/2
-f(u, t) = -r*(u - Ta)
-v(t) = Ta + (T0 - Ta) * exp(-r*t)
-p = plot(v, 0, 6, linewidth=4, legend=false)
-[plot!(p, x -> v(a) + f(v(a), a) * (x-a), 0, 6) for a in 1:2:5]
-p
+```julia; echo=false
+let
+    T0, Ta, r = 200, 72, 1/2
+    f(u, t) = -r*(u - Ta)
+    v(t) = Ta + (T0 - Ta) * exp(-r*t)
+    p = plot(v, 0, 6, linewidth=4, legend=false)
+    [plot!(p, x -> v(a) + f(v(a), a) * (x-a), 0, 6) for a in 1:2:5]
+    p
+end
 ```
 
 
@@ -725,10 +730,8 @@ plot!(f,  linewidth=5)
 
 In general, if the first-order equation is written as $y'(x) = F(y,x)$, then we plot a "function" that takes $(x,y)$ and returns an $x$ value of $1$ and a $y$ value of $F(y,x)$, so the slope is $F(y,x)$.
 
-```julia; echo=false
-note(L"""The order of variables in $F(y,x)$ is conventional with the equation $y'(x) = F(y(x),x)$.
-""")
-```
+!!! note
+	The order of variables in $F(y,x)$ is conventional with the equation $y'(x) = F(y(x),x)$.
 
 
 The plots are also useful for illustrating solutions for different initial conditions:
@@ -780,8 +783,8 @@ choices = [
 "``[-1, 4]``",
 "``[-1, 0]``",
 "``[1-\\sqrt{5}, 1 + \\sqrt{5}]``"]
-ans = 4
-radioq(choices, ans)
+answ = 4
+radioq(choices, answ)
 ```
 
 
@@ -891,8 +894,8 @@ eqn = diff(x^2*D(u)(x), x)
 out = dsolve(eqn, u(x), ics=Dict(u(1)=>2, u(10) => 1)) |> rhs
 out(5)  # 10/9
 choices = ["``10/9``", "``3/2``", "``9/10``", "``8/9``"]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 
@@ -909,6 +912,6 @@ choices = [
 "The limit does not exist, but the limit to `oo` gives a quadratic polynomial in `x`, mirroring the first part of that example.",
 "The limit does not exist -- there is a singularity -- as seen by setting `gamma=0`."
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```