pull examples and complete homework 1

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+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+
+
+# INITIAL CONDITIONALS AND PARAMETERS
+a = 10.0 # birth of new members
+b = 3.0 # death of members
+n0 = 10.0 # initial number of members
+t_final = 1.0 # final time of simulation
+p = 0.0 # parameters (not used here)
+dt = 0.01 # time step for euler's
+
+
+# EULER'S METHOD -> APPROXIMATE ANSWER
+steps = Int64(t_final/dt) # number of time steps
+
+n = zeros(steps+1) # initial array of members
+v = zeros(steps+1) # initial array of rate of change of members
+t = zeros(steps+1) # initial array of time intervals
+
+function dynamics!(n::Vector{Float64}, v::Vector{Float64}, t::Vector{Float64}) 
+    for i in 1:steps
+        # equation: dn = dt(aN - bN^2)
+        dn = dt*(a*n[i] - b*n[i]*n[i])
+        vn = dt*(a-2.0*b*n[i])
+
+        n[i+1] = n[i] + dn
+        v[i+1] = v[i] + vn
+        t[i+1] = t[i] + dt
+    end
+end
+
+# calcuate with current dt, store into arrays
+n[1] = n0
+v[1] = 0.0
+t[1] = 0.0
+dynamics!(n, v, t)
+
+
+# USING ODE SOLVER -> EXACT ANSWER
+
+function tendency!(dnv::Vector{Float64}, nv::Vector{Float64}, p, t::Float64) # ! notation tells us that arguments will be modified
+   n = nv[1] # 2D phase space; use vcat(x, v) to combine 2 vectors
+   v = nv[2] # dn/dt = v
+
+   dnv[1] = a*n - b*n*n
+   dnv[2] = a - 2.0*b*n
+end
+
+i0 = [n0, v0] # set initial conditions 
+tspan = (0.0, t_final) # span of time to simulate
+prob = ODEProblem(tendency!, i0, tspan, p) # specify ODE
+sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) # solve using Tsit5 algorithm to specified accuracy
+n_exact = sol[1, :] # extract the population values over time
+
+
+# PLOTTING AND COMPARISON
+println("Parameters (a, b, n0, t_final): ", a, ", ", b, ", ", n0, ", ", t_final)
+println("Final population (at ", t_final, ") via Euler's Method:\t", n[end])
+println("Final population (at ", t_final, ") via ODE Solver:\t", n_exact[end])
+
+plot_title = "Population v. time (w/ n0,a,b= " * string(n0) * ", " * string(a) * ", " * string(b) * ")"
+plot(t, n, label="Euler's Method (dt = .01)", title=plot_title, lw=2, xlabel="time", ylabel="population")
+plot!(sol.t, n_exact, label="Exact Solution (ode solver)", lw=2) # plot!() to add to existing plot
+
+# NOTE: uncomment the two lines below if you also want to plot the next derativate
+# plot!(t, v, label="d^2n/dt^2 (Euler's Method)", lw=2)
+# plot!(sol.t, sol[2, :], label="d^2n/dt^2(ode solver)", lw=2)
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