︠a2b01e63-ba6e-4151-b920-f30504f3d42fi︠
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%html
An interact is a small applet that allows you to change the parameters and see results.  This interact solves the differential equation modeling a spring system which has no forcing function.
︡f59381ab-98db-431e-8e44-89d0bb31fc3a︡{"html": "<font color='black'>An interact is a small applet that allows you to change the parameters and see results.  This interact solves the differential equation modeling a spring system which has no forcing function.</font>"}︡
︠39ba355d-79a1-4050-a4cb-efabae7199bfi︠
%hide
#auto
RF=RealField(10)
var("t")
t0=float(0)
︡d8f407f0-f3de-44aa-9a42-b7521c29e9ce︡{"stdout": "CPU time: 0.00 s,  Wall time: 0.00 s"}︡
︠b0c57eb8-589f-4587-ab6d-82bb3cac96e2i︠
%hide
%cython
#auto
#from math import exp

cdef extern from "math.h":
    double exp(double x)
    double cos(double x)
    double sin(double x)
    double atan2(double y, double x)

cpdef list solve_diffeq(double a, double b, double c, double t0, double y0, double y0p):
    """
    Returns a list starting with discriminant
    if discriminant > 0, then the list is [disc, r1, r2, c1, c2]
    if discriminant == 0, then the list is [disc,r1,c1,c2]
    if discriminant < 0, then the list is [disc, lambda, mu, c1, c2,r,delta]
    """
    cdef list result = []
    cdef double discriminant = b*b-4*a*c
    cdef double const[2]
    result.append(discriminant)
    cdef double r1, r2, pos_disc_root, y1t0, y1difft0, y2t0, y2difft0, denominator, c1, c2
    cdef double lam, mu, elam, sinmu, cosmu, r, delta
    if discriminant > 0:
        pos_disc_root = sqrt(discriminant)
        r1 = (-b + pos_disc_root)/(2*a)
        r2 = (-b - pos_disc_root)/(2*a)
        result.append(r1)
        result.append(r2)
        y1t0 = exp(r1*t0)
        y1difft0 = r1*y1t0
        y2t0 = exp(r2*t0)
        y2difft0 = r2*y2t0
        denominator = y1t0*y2difft0 - y2t0*y1difft0
        c1 =  (y0*y2difft0 - y0p*y2t0)/denominator
        c2 = (-y0*y1difft0 + y0p*y1t0)/denominator    
        result.append(c1)
        result.append(c2)
    elif discriminant == 0:
        r1 = -b/(2*a)
        result.append(r1)

        y1t0 = exp(r1*t0)
        y1difft0 = r1*exp(r1*t0)
        y2t0 = t0*y1t0
        y2difft0 = y1t0 + r1*y2t0
        denominator = y1t0*y2difft0 - y2t0*y1difft0
        c1 =  (y0*y2difft0 - y0p*y2t0)/denominator
        c2 = (-y0*y1difft0 + y0p*y1t0)/denominator    
        result.append(c1)
        result.append(c2)
    else:
        lam = -b/(2*a)
        mu = sqrt(-discriminant)/(2*a)
        result.append(lam)
        result.append(mu)
        elam = exp(lam*t0)
        cosmu = cos(mu*t0)
        sinmu = sin(mu*t0)    
        y1t0 = elam*cosmu
        y1difft0 = elam*(lam*cosmu-mu*sinmu)
        y2t0 = elam*sinmu
        y2difft0 = elam*(lam*sinmu+mu*cosmu)
        denominator = y1t0*y2difft0 - y2t0*y1difft0
        c1 =  (y0*y2difft0 - y0p*y2t0)/denominator
        c2 = (-y0*y1difft0 + y0p*y1t0)/denominator    
        result.append(c1)
        result.append(c2)
        r = sqrt(c1*c1+c2*c2)
        delta = atan2(c2,c1)
        result.append(r)
        result.append(delta)
        
    return result
︡7ab98f90-5ca6-430a-8391-db13b1c84b36︡{"stdout": "CPU time: 0.04 s,  Wall time: 4.25 s"}︡
︠5386d969-7ba4-400b-89b4-b519e707c62ai︠
%hide
#auto
@interact
def _(initial=input_grid(1,3,default=[-1/12,2,5],label="y(0), y'(0),t_max",width=10),
    abc=input_grid(1,3,default=[3/32,0,12],label="m,gamma,k", width=10)):
    y0,y0p,tmax = initial[0]
    a,b,c = abc[0]
    plots = []
    a = float(a)
    b = float(b)
    c = float(c)
    soln_nums = solve_diffeq(a,b,c,t0,y0,y0p)
    discriminant = soln_nums[0]

    if discriminant > 0:
        r1,r2,c1,c2 = soln_nums[1:]
        soln = c1*e^(r1*t)+c2*e^(r2*t)
        linear_combo = ["%se^{%st}"%(RF(constant),RF(rval)) for constant,rval in ((c1,r1),(c2,r2)) if constant !=0]
        soln_n = "$y(t) = " + '+'.join(linear_combo) + "$"
    elif discriminant == 0:
        r1,c1,c2 = soln_nums[1:]
        soln = (c1+c2*t)*e^(r1*t)
        linear_combo = ["%s%s"%(RF(constant),y) for constant,y in ((c1,"e^{%st}"%RF(r1)),(c2,"te^{%st}"%RF(r1))) if constant !=0]
        soln_n = "$y(t) = " + '+'.join(linear_combo) + "$"

    else:
        lam,mu,c1,c2,R,delta = soln_nums[1:]
        soln = e^(lam*t)*(c1*cos(mu*t)+c2*sin(mu*t))
        linear_combo = ["%s%s"%(RF(constant),y) for constant,y in ((c1,"\cos(%st)"%RF(mu)),(c2,"\sin(%st)"%RF(mu))) if constant !=0]
        soln_n = "$y(t) = e^{%st}("%RF(lam) + '+'.join(linear_combo) + ")$"
        soln2 = R*e^(lam*t)*cos(mu*t-delta)
        soln2_n = "%se^{%st}\cos(%st-%s)"%(RF(R),RF(lam),RF(mu),RF(delta))
        plots.extend([plot(R*e^(lam*t),(t,0,tmax),rgbcolor=(0.1,0.1,0.1)),
            plot(-R*e^(lam*t),(t,0,tmax),rgbcolor=(0.1,0.1,0.1))])

    line1 = "$\\left(%s \\right)y''+\\left(%s \\right)y'+\\left(%s \\right)y=0; \quad$"%(latex(a),latex(b),latex(c))
    line1 += "$y(%s)=%s,\quad y'(%s)=%s$"%(latex(t0), latex(y0), latex(t0), latex(y0p))
    line1 += "$\quad$ discriminant = $%s$"%latex(discriminant)
    line2 = soln_n
    if discriminant < 0:
        line2 += "$\quad = \quad "+soln2_n+"$"
    html(line1)
    html(line2)
    plots.append(plot(soln,(t,0,tmax)))
    show(sum(plots))
︡18cea596-780c-49b0-8a36-e4b740e684a9︡{"html": "<div padding=6 id='div-interact-2'> <table width=800px height=20px bgcolor='#c5c5c5'\n                 cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color=\"black\">y(0), y'(0),t_max&nbsp;</font></td><td><table><tr><td><input type='text' value='-1/12' size='10' onchange='interact(2, \"sage.server.notebook.interact.update(2, \\\"initial\\\", 3, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64( \"[[\"+jQuery(this).parents(\"table\").eq(0).find(\"tr\").map(function(){return jQuery(this).find(\"input\").map(function() {return jQuery(this).val();}).get().join(\",\");}).get().join(\"],[\")+\"]]\" )+\"\\\"), globals())\")'></input></td><td><input type='text' value='2' size='10' onchange='interact(2, \"sage.server.notebook.interact.update(2, \\\"initial\\\", 3, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64( \"[[\"+jQuery(this).parents(\"table\").eq(0).find(\"tr\").map(function(){return jQuery(this).find(\"input\").map(function() {return jQuery(this).val();}).get().join(\",\");}).get().join(\"],[\")+\"]]\" )+\"\\\"), globals())\")'></input></td><td><input type='text' value='5' size='10' onchange='interact(2, \"sage.server.notebook.interact.update(2, \\\"initial\\\", 3, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64( \"[[\"+jQuery(this).parents(\"table\").eq(0).find(\"tr\").map(function(){return jQuery(this).find(\"input\").map(function() {return jQuery(this).val();}).get().join(\",\");}).get().join(\"],[\")+\"]]\" )+\"\\\"), globals())\")'></input></td></tr></table></td></tr>\n<tr><td align=right><font color=\"black\">m,gamma,k&nbsp;</font></td><td><table><tr><td><input type='text' value='3/32' size='10' onchange='interact(2, \"sage.server.notebook.interact.update(2, \\\"abc\\\", 4, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64( \"[[\"+jQuery(this).parents(\"table\").eq(0).find(\"tr\").map(function(){return jQuery(this).find(\"input\").map(function() {return jQuery(this).val();}).get().join(\",\");}).get().join(\"],[\")+\"]]\" )+\"\\\"), globals())\")'></input></td><td><input type='text' value='0' size='10' onchange='interact(2, \"sage.server.notebook.interact.update(2, \\\"abc\\\", 4, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64( \"[[\"+jQuery(this).parents(\"table\").eq(0).find(\"tr\").map(function(){return jQuery(this).find(\"input\").map(function() {return jQuery(this).val();}).get().join(\",\");}).get().join(\"],[\")+\"]]\" )+\"\\\"), globals())\")'></input></td><td><input type='text' value='12' size='10' onchange='interact(2, \"sage.server.notebook.interact.update(2, \\\"abc\\\", 4, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64( \"[[\"+jQuery(this).parents(\"table\").eq(0).find(\"tr\").map(function(){return jQuery(this).find(\"input\").map(function() {return jQuery(this).val();}).get().join(\",\");}).get().join(\"],[\")+\"]]\" )+\"\\\"), globals())\")'></input></td></tr></table></td></tr>\n</table><div id='cell-interact-2'><?__SAGE__START>\n        <table border=0 bgcolor='#white' width=100% height=100%>\n        <tr><td bgcolor=white align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>\n        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>\n        </table><?__SAGE__END></div></td>\n                 </tr></table></div>\n                 "}︡{"stdout": "CPU time: 0.00 s,  Wall time: 0.00 s"}︡

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