MATH383

Overview

Example. $y^{''} - 5 y^{'} + 6 y = 0$ , $y (0) = 1$ , $y^{'} (0) = 0$ , $r^{2} - 5 r + 6 = 0 \Rightarrow r_{1} = 2, r_{3} = 3$

y (t) = c_{1} e^{2 t} + c_{2} e^{3 t}, y (0) = c_{1} + c_{2} = 1, y^{'} (0) = 2 c_{1} + 3 c_{2} = 0 .

(%i1)

ode: 'diff(y,t,2)-5*'diff(y,t)+6*y

$(%o1) \frac{d^{2} y}{d t^{2}} - 5 (\frac{d y}{d t}) + 6 y$

(%i2)

sol: ode2(ode,y,t)

$(%o2) y = % k 1 e^{3 t} + % k 2 e^{2 t}$

(%i3)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o3) y = 3 e^{2 t} - 2 e^{3 t}$

(%i4)

Example. $y^{''} + 2 y^{'} + 2 y = 0$ , $y (0) = 1$ , $y^{'} (0) = 0$ , $r^{2} + 2 r + 2 = 0 \Rightarrow r_{1, 2} = 1 \pm i$

α = 1, β = 1, y (t) = e^{- α t} (c_{1} \cos β t + c_{2} \sin β t) .

(%i4)

ode: 'diff(y,t,2)+2*'diff(y,t)+2*y

$(%o4) \frac{d^{2} y}{d t^{2}} + 2 (\frac{d y}{d t}) + 2 y$

(%i5)

sol: ode2(ode,y,t)

$(%o5) y = e^{- t} (% k 1 \sin (t) + % k 2 \cos (t))$

(%i7)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o7) y = e^{- t} (\sin (t) + \cos (t))$

(%i8)

Example. $y^{''} + 2 y^{'} + y = 0$ , $y (0) = 1$ , $y^{'} (0) = 0$ , $r^{2} + 2 r + 1 = 0 \Rightarrow r_{} = - 1$

y (t) = e^{- r t} (c_{1} + c_{2} t) .

(%i8)

ode: 'diff(y,t,2)+2*'diff(y,t)+y

$(%o8) \frac{d^{2} y}{d t^{2}} + 2 (\frac{d y}{d t}) + y$

(%i9)

sol: ode2(ode,y,t)

$(%o9) y = (% k 2 t + % k 1) e^{- t}$

(%i10)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o10) y = (t + 1) e^{- t}$

(%i11)

$m y^{''} + k y = f (t)$ , $m r^{2} + k = 0$ , $r_{1, 2} = \pm i ω$ , $ω = \sqrt{k / m}$
Non-resonant forcing: $f (t)$ not proportional to $\cos (ω t), \sin (ω t)$
Steps:
1. Solve homogeneous equation $y_{h} = a \cos (ω t) + b \sin (ω t)$
2. Search for a particular solution suggested by force term
(%i14)

ode: 'diff(y,t,2)+4*y-t
$(%o14) \frac{d^{2} y}{d t^{2}} + 4 y - t$
(%i15)

sol: ode2(ode,y,t)
$(%o15) y = % k 1 \sin (2 t) + % k 2 \cos (2 t) + \frac{t}{4}$
(%i16)

ic2(sol,t=0,y=1,'diff(y,t)=0)
$(%o16) y = - \frac{\sin (2 t)}{8} + \cos (2 t) + \frac{t}{4}$
(%i17)

$m y^{''} + k y = f (t)$ , $m r^{2} + k = 0$ , $r_{1, 2} = \pm i ω$ , $ω = \sqrt{k / m}$
Non-resonant forcing: $f (t)$ proportional to $\cos (ω t), \sin (ω t)$
Steps:
1. Solve homogeneous equation $y_{h} = a \cos (ω t) + b \sin (ω t)$
2. Search for a particular solution of form $t (A \cos ω t + B \sin ω t)$
(%i17)

ode: 'diff(y,t,2)+4*y-sin(2*t)
$(%o17) \frac{d^{2} y}{d t^{2}} + 4 y - \sin (2 t)$
(%i18)

sol: ode2(ode,y,t)
$(%o18) y = % k 1 \sin (2 t) - \frac{t \cos (2 t)}{4} + % k 2 \cos (2 t)$
(%i13)

ic2(sol,t=0,y=1,'diff(y,t)=0)
$(%o13) y = - \frac{17 e^{3 t}}{9} + \frac{11 e^{2 t}}{4} + \frac{5 + 6 t}{36}$
(%i14)

Example. $y^{''} - 5 y^{'} + 6 y = \sin t$ , $r^{2} - 5 r + 6 = 0 \Rightarrow r_{1} = 2, r_{3} = 3$

y_{h} (t) = c_{1} e^{2 t} + c_{2} e^{3 t}, y_{p} = A \sin t + B \cos t .

(%i19)

ode: 'diff(y,t,2)-5*'diff(y,t)+6*y - sin(t)

$(%o19) \frac{d^{2} y}{d t^{2}} - 5 (\frac{d y}{d t}) + 6 y - \sin (t)$

(%i20)

sol: ode2(ode,y,t)

$(%o20) y = \frac{\cos (t) + \sin (t)}{10} + % k 1 e^{3 t} + % k 2 e^{2 t}$

(%i21)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o21) y = \frac{\cos (t) + \sin (t)}{10} - \frac{19 e^{3 t}}{10} + \frac{14 e^{2 t}}{5}$

(%i22)

Example. $y^{''} - 5 y^{'} + 6 y = e^{2 t}$ , $r^{2} - 5 r + 6 = 0 \Rightarrow r_{1} = 2, r_{3} = 3$

y_{h} (t) = c_{1} e^{2 t} + c_{2} e^{3 t}, y_{p} = A e^{2 t} + B t e^{2 t} .

(%i22)

ode: 'diff(y,t,2)-5*'diff(y,t)+6*y - exp(2*t)

$(%o22) \frac{d^{2} y}{d t^{2}} - 5 (\frac{d y}{d t}) + 6 y - e^{2 t}$

(%i23)

sol: ode2(ode,y,t)

$(%o23) y = % k 1 e^{3 t} + (- 1 - t) e^{2 t} + % k 2 e^{2 t}$

(%i24)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o24) y = - e^{3 t} + (- 1 - t) e^{2 t} + 3 e^{2 t}$

(%i25)

Example. $y^{''} + 2 y^{'} + 2 y = t$ , $r^{2} + 2 r + 2 = 0 \Rightarrow r_{1, 2} = 1 \pm i$

α = 1, β = 1, y_{h} (t) = e^{- α t} (c_{1} \cos β t + c_{2} \sin β t), y_{p} = A + B t

(%i25)

ode: 'diff(y,t,2)+2*'diff(y,t)+2*y -t

$(%o25) \frac{d^{2} y}{d t^{2}} + 2 (\frac{d y}{d t}) + 2 y - t$

(%i26)

sol: ode2(ode,y,t)

$(%o26) y = e^{- t} (% k 2 \cos (t) + % k 1 \sin (t)) + \frac{t - 1}{2}$

(%i27)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o27) y = e^{- t} (\frac{3 \cos (t)}{2} + \sin (t)) + \frac{t - 1}{2}$

(%i28)

Example. $y^{''} + 2 y^{'} + 2 y = e^{- t} \sin t$ , $r^{2} + 2 r + 2 = 0 \Rightarrow r_{1, 2} = 1 \pm i$

α = 1, β = 1, y_{h} (t) = e^{- α t} (c_{1} \cos β t + c_{2} \sin β t), y_{p} = t e^{- t} (A \cos t + B \sin t)

(%i31)

ode: 'diff(y,t,2)+2*'diff(y,t)+2*y - exp(-t)*sin(t)

$(%o31) \frac{d^{2} y}{d t^{2}} + 2 (\frac{d y}{d t}) + 2 y - e^{- t} \sin (t)$

(%i32)

sol: ode2(ode,y,t)

$(%o32) y = e^{- t} (% k 2 \cos (t) + % k 1 \sin (t)) - \frac{t e^{- t} \cos (t)}{2}$

(%i39)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o39) y = e^{- t} (\cos (t) + \frac{3 \sin (t)}{2}) - \frac{t e^{- t} \cos (t)}{2}$

(%i40)

Example. $y^{''} + 2 y^{'} + y = t$ , $r^{2} + 2 r + 1 = 0 \Rightarrow r_{} = - 1$

y_{h} (t) = e^{- r t} (c_{1} + c_{2} t), y_{p} = A + B t

(%i40)

ode: 'diff(y,t,2)+2*'diff(y,t)+y - t

$(%o40) \frac{d^{2} y}{d t^{2}} + 2 (\frac{d y}{d t}) + y - t$

(%i41)

sol: ode2(ode,y,t)

$(%o41) y = (% k 1 + % k 2 t) e^{- t} + t - 2$

(%i42)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o42) y = (3 + 2 t) e^{- t} + t - 2$

(%i43)

Example. $y^{''} + 2 y^{'} + y = t e^{- t}$ , $r^{2} + 2 r + 1 = 0 \Rightarrow r_{} = - 1$

y_{h} (t) = e^{- r t} (c_{1} + c_{2} t), y_{p} = (A + B t) t^{2} e^{- r t}

(%i43)

ode: 'diff(y,t,2)+2*'diff(y,t)+y - t*exp(-t)

$(%o43) \frac{d^{2} y}{d t^{2}} + 2 (\frac{d y}{d t}) + y - t e^{- t}$

(%i44)

sol: ode2(ode,y,t)

$(%o44) y = \frac{t^{3} e^{- t}}{6} + (% k 1 + % k 2 t) e^{- t}$

(%i45)

ic2(sol,t=0,y=1,'diff(y,t)=0)

$(%o45) y = \frac{t^{3} e^{- t}}{6} + (1 + t) e^{- t}$

(%i46)