352 - Topic 11 - Linear Constant Coefficient DEs with operators -- Sage

var('a,t') y = e^(a*t) pretty_print(html("$$D[%s]$$"%(latex(y)) )) show(diff(y,t))

$D[e^{\left(a t\right)}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}a e^{\left(a t\right)}$

reset() var('a,t,n') assume(n>0) assume(n, 'integer') y = e^(a*t) pretty_print(html("$$D^n[%s]$$"%(latex(y)) )) for k in range(2,10): y = diff(e^(a*t),t,k) pretty_print(html("$$D^{%s}"%str(k)+"[e^{at}] = %s$$"%(latex(y)) ))

$D^n[e^{\left(a t\right)}]$

$D^{2}[e^{at}] = a^{2} e^{\left(a t\right)}$

$D^{3}[e^{at}] = a^{3} e^{\left(a t\right)}$

$D^{4}[e^{at}] = a^{4} e^{\left(a t\right)}$

$D^{5}[e^{at}] = a^{5} e^{\left(a t\right)}$

$D^{6}[e^{at}] = a^{6} e^{\left(a t\right)}$

$D^{7}[e^{at}] = a^{7} e^{\left(a t\right)}$

$D^{8}[e^{at}] = a^{8} e^{\left(a t\right)}$

$D^{9}[e^{at}] = a^{9} e^{\left(a t\right)}$

var('a,t') y = e^(a*t) pretty_print(html("$$(D-a)[%s]$$"%(latex(y)) )) show(diff(y,t) - a*y )

$(D-a)[e^{\left(a t\right)}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

var('a,t') y = function('y')(t) pretty_print(html("$$(D-a)[e^{at}%s]$$"%(latex(y)) )) show(diff(e^(a*t)*y,t) - a*e^(a*t)*y )

$(D-a)[e^{at}y\left(t\right)]$

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(a t\right)} \frac{\partial}{\partial t}y\left(t\right)$

var('a,t') y = function('y')(t) Y = e^(a*t)*y for k in range(1,5): pretty_print(html("$$(D-a)^%s"%str(k)+"[e^{at}%s]$$"%(latex(y)) )) Y = diff(Y,t) - a*Y show(Y)

$(D-a)^1[e^{at}y\left(t\right)]$

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(a t\right)} \frac{\partial}{\partial t}y\left(t\right)$

$(D-a)^2[e^{at}y\left(t\right)]$

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(a t\right)} \frac{\partial^{2}}{(\partial t)^{2}}y\left(t\right)$

$(D-a)^3[e^{at}y\left(t\right)]$

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(a t\right)} \frac{\partial^{3}}{(\partial t)^{3}}y\left(t\right)$

$(D-a)^4[e^{at}y\left(t\right)]$

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(a t\right)} \frac{\partial^{4}}{(\partial t)^{4}}y\left(t\right)$

var('a,t') n=10 for r in range(n+1): # all but the last should be 0 Y = e^(a*t)*t^r for k in range(n): # take n derivatives Y = diff(Y,t) - a*Y pretty_print(html("$$(D-a)^%s"%str(n)+"[e^{at}t^{%s}]$$"%(r) )) show(Y)

$(D-a)^10[e^{at}t^{0}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{1}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{2}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{3}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{4}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{5}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{6}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{7}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{8}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{9}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$(D-a)^10[e^{at}t^{10}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}3628800 \, e^{\left(a t\right)}$

var('b,c,t') pretty_print(html("Solve $$(D^2 + 2bD + c^2 + b^2)[y]=0$$")) pretty_print(html("$$(D^2 + 2b + c^2 + b^2)[%s]$$"%(latex(y)) )) result = diff(y,t,2) + 2*b*diff(y,t) +(c^2+b^2)*y show(result.simplify_full())

Solve $(D^2 + 2bD + c^2 + b^2)[y]=0$

$(D^2 + 2b + c^2 + b^2)[a^{9} e^{\left(a t\right)}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(a^{11} + 2 \, a^{10} b + a^{9} b^{2} + a^{9} c^{2}\right)} e^{\left(a t\right)}$

Solve

var('b,c,t') y = e^(-b*t)*cos(c*t) pretty_print(html("$$(D^2 + 2bD + c^2 + b^2)[%s]$$"%(latex(y)) )) result = diff(y,t,2) + 2*b*diff(y,t) +(c^2+b^2)*y show(result.simplify_full())

$(D^2 + 2bD + c^2 + b^2)[\cos\left(c t\right) e^{\left(-b t\right)}]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

var('b,c,t') y = e^(-b*t)*sin(c*t) pretty_print(html("$$(D^2 + 2bD + c^2 + b^2)[%s]$$"%(latex(y)) )) result = diff(y,t,2) + 2*b*diff(y,t) +(c^2+b^2)*y show(result.simplify_full())

$(D^2 + 2bD + c^2 + b^2)[e^{\left(-b t\right)} \sin\left(c t\right)]$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

# page 184 of text auxilary = x^3 + 5*x^2 + 6*x + 2 auxilary.factor().show() solve(auxilary,x)

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x^{2} + 4 \, x + 2\right)} {\left(x + 1\right)}$

[x == -sqrt(2) - 2, x == sqrt(2) - 2, x == -1]

[x == -sqrt(2) - 2, x == sqrt(2) - 2, x == -1]

var('t') # first root y = exp(-t) DE = diff(y,t,3) + 5*diff(y,t,2) + 6*diff(y,t) + 2*y show("y =",y) show(DE) pretty_print("---------") # second root y = exp((sqrt(2)-2)*t) DE = diff(y,t,3) + 5*diff(y,t,2) + 6*diff(y,t) + 2*y show("y =",y) show(DE) DE=(DE/e^(t*(sqrt(2)-2))).expand() show(DE) pretty_print("---------") # third root y = exp((-sqrt(2)-2)*t) DE = diff(y,t,3) + 5*diff(y,t,2) + 6*diff(y,t) + 2*y show("y =",y) show(DE) DE=(DE/e^(t*(sqrt(2)-2))).expand() show(DE)

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|y|\phantom{\verb!x!}\verb|=| e^{\left(-t\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|---------|$

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|y|\phantom{\verb!x!}\verb|=| e^{\left(t {\left(\sqrt{2} - 2\right)}\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\sqrt{2} - 2\right)}^{3} e^{\left(t {\left(\sqrt{2} - 2\right)}\right)} + 5 \, {\left(\sqrt{2} - 2\right)}^{2} e^{\left(t {\left(\sqrt{2} - 2\right)}\right)} + 6 \, {\left(\sqrt{2} - 2\right)} e^{\left(t {\left(\sqrt{2} - 2\right)}\right)} + 2 \, e^{\left(t {\left(\sqrt{2} - 2\right)}\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|---------|$

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|y|\phantom{\verb!x!}\verb|=| e^{\left(-t {\left(\sqrt{2} + 2\right)}\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(\sqrt{2} + 2\right)}^{3} e^{\left(-t {\left(\sqrt{2} + 2\right)}\right)} + 5 \, {\left(\sqrt{2} + 2\right)}^{2} e^{\left(-t {\left(\sqrt{2} + 2\right)}\right)} - 6 \, {\left(\sqrt{2} + 2\right)} e^{\left(-t {\left(\sqrt{2} + 2\right)}\right)} + 2 \, e^{\left(-t {\left(\sqrt{2} + 2\right)}\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

# Consider an easy second order with imaginary roots var('t') pretty_print("D^2 + 4") TBA # first root y = exp(-t) DE = diff(y,t,3) + 5*diff(y,t,2) + 6*diff(y,t) + 2*y show("y =",y) show(DE) # second root y = exp((-sqrt(2)-2)*t) DE = diff(y,t,3) + 5*diff(y,t,2) + 6*diff(y,t) + 2*y show("y =",y.expand()) show(DE) # third root

# Let's try a fourth order with complex roots var('t') # linear coefficient = b # constant term = c^2 + b^2 y = sin(t) # y = cos(t) # y = e^(t) # y = e^(-t) DE = diff(y,t,4) - y show("y =",y) show(DE)

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|y|\phantom{\verb!x!}\verb|=| \sin\left(t\right)$

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

var('t') pretty_print("(D^2+8D+20)[y]=0") y1 = e^(-4*t)*cos(2*t) pretty_print(y1) y2 = e^(-4*t)*sin(2*t) pretty_print(y2) diff(y1,t,2)+8*diff(y1,t)+20*y1 diff(y2,t,2)+8*diff(y2,t)+20*y2

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|(D^2+8D+20)[y]=0|$

$\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(2 \, t\right) e^{\left(-4 \, t\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-4 \, t\right)} \sin\left(2 \, t\right)$

352 - Topic 11 - Linear Constant Coefficient DEs with operators

10 hours ago by Professor352