---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------Second-order differential equations# It is convenient to useWolframAlphato find the solutions, and thenRto plot their # charts. In any case, withR Ican check the solutions. # # A car that is moving with a constant acceleration of 5 m/s^2 and has a speed of 1 m/s # at a time when it is 2 m from a reference point. # WithWolframAlpha: f"(x)=5, f'(0)=1, f(0)=2 # I obtain: f(x) = 5*x^2/2+x+2. The graph withR: f = function(x) 5*x^2/2+x+2; graphF(f, 0,20, "brown") # The check df = function(x) eval(deriv(f,"x")) deriv2(f,"x"); df(0); f(0) # 5 1 2 # # Second example. x = time, y = position of an object of mass m subjected to force # -k*y by a spring, with k = 4, m = 1. F = m*y"(t) -> y"(x) = -4/1*y(x). # WithWolframAlpha. I obtain y(x) = c2*sin(2*x)+c1*cos(2*x). # If I put y"(x) = -4*y(x), y(0)=2, y'(0)=0 I obtain: y(x) = 2*cos(2*x) and the graphs: # The check withRf = function(x) 2*cos(2*x) df = function(x) eval(deriv(f,"x")); d2f = function(x) eval(deriv2(f,"x")) F = function(x) d2f(x)+4*f(x); x=1:5; F(x) # 0 0 0 0 0 f(0); df(0) # 2 0 # The graph withR: graph2F(f, 0,20, "brown") # Or: TICKx=pi/2; TICKy=1/2; Plane2(0,20, -2,2) graph2(f, 0,20, "brown") underY(-1,-1); underY(1,1); underY(-2,-2); underY(2,2); underY(0,0) underX(bquote(2*pi),2*pi); underX(bquote(4*pi),4*pi); underX(bquote(6*pi),6*pi) underX(0,0) # # How to trace thephase curve, ie the curve (y(x),y'(x)) [the relationship between # "y" and its "speed"]? df = function(x) eval(deriv(f,"x")) # Where does df change? x=seq(0,20,0.1); c(min(df(x)),max(df(x)))#-3.999647 3.999961 Plane(-2,2, -4,4) param(f,df, 0,20, "brown") # In this case, as a phase diagram I get an ellipse: the solution isperiodic# # Third example.Forced oscillations, with external oscillating force F·cos(Ω·t): # y" = -(k/m)·y + F/m·cos(Ω·t). Consider the following case: F=1, Ω=2 # and the other data as above: y"(x)=-4*y(x)+cos(2*x), y(0)=2, y'(0)=0 # With WolframAlpha I obtain 2*cos(2*x)+1/4*x*sin(2*x) and the graphs: # The check withRf = function(x) 2*cos(2*x)+1/4*x*sin(2*x) df = function(x) eval(deriv(f,"x")); d2f = function(x) eval(deriv2(f,"x")) F = function(x) d2f(x)+4*f(x)-cos(2*x); x=1:5; F(x) # -6.106227e-16 1.110223e-16 3.330669e-16 1.110223e-16 -1.110223e-16 Practically 0 f(0); df(0) # 2 0 # The graph: graph2F(f, 0,30, "brown")# Thephase curvedf = function(x) eval(deriv(f,"x")) # Where does df change? x=seq(0,30,0.1); c(min(df(x)),max(df(x)))#-15.30207 14.49992 Plane(-8,8, -15,15); param(f,df, 0,30, "brown")# For these values of F and Ω there is the phenomenon ofresonance(oscillations # increase in amplitude). # # Fourth example.Damped oscillations, with y" = -(k/m)·y - k1·y' (with frictional # force proportional to speed). Example with k1 = 1 # y"(x) = -4*y(x)-y'(x), y(0)=1, y'(0)=3 # With WolframAlpha:: # y(x) = 1/15*exp(-x/2)*(7*sqrt(15)*sin((sqrt(15)*x)/2)+15*cos((sqrt(15)*x)/2)) # WithR: S = sqrt(15) f = function(x) 1/15*exp(-x/2)*(7*S*sin((S*x)/2)+15*cos((S*x)/2)) graph2F(f, 0,20, "brown") df = function(x) eval(deriv(f,"x")) x=seq(0, 20, 0.1); c(min(df(x)),max(df(x)))#-2.307363 3.000000 Plane(-1,2, -3,3); para(f,df, 0,20, "brown")Other examples of use