Sat Mar 14 16:31:42 MST 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Ordinary Difference and Differential Equations\ > ---------- *) 0. Second In[6]:= (* Second order linear recurrence equationn: r(n) = (n - 1)^2 + m n *) 0. Second In[7]:= << DiscreteMath`RSolve` 1.61 Second In[8]:= r[n + 2] - 2 * r[n + 1] + r[n] == 2 0. Second Out[8]= r[n] - 2 r[1 + n] + r[2 + n] == 2 In[9]:= Simplify[RSolve[{%, r[0] == 1, r[1] == m}, r[n], n]] 0.46 Second 2 Out[9]= {{r[n] -> 1 + (-2 + m) n + n }} In[10]:= (* => r(n) = 3^n - 2^n [Cohen, p. 67] *) 0. Second In[11]:= RSolve[{r[n] == 5*r[n - 1] - 6*r[n - 2], r[0] == 0, r[1] == 1},\ > r[n], n] 0.21 Second n n Out[11]= {{r[n] -> -2 + 3 }} In[12]:= (* => r(n) = Fibonacci[n + 1] [Cohen, p. 83] *) 0. Second In[13]:= RSolve[{r[n] == r[n - 1] + r[n - 2], r[1] == 1, r[2] == 2}, r[n], n] 1.11 Second Out[13]= {{r[n] -> 1 Sqrt[5] 1 Sqrt[5] -1 + n 1 - Sqrt[5] (-(-) - -------) (- + -------) (-1 + -----------) 2 2 2 2 2 > ------------------------------------------------------- - Sqrt[5] 1 Sqrt[5] -1 + n 1 Sqrt[5] 1 + Sqrt[5] (- - -------) (-(-) + -------) (-1 + -----------) 2 2 2 2 2 > -------------------------------------------------------}} Sqrt[5] In[14]:= FullSimplify[%] 3.55 Second Out[14]= {{r[n] -> 1 Sqrt[5] n 1 + Sqrt[5] n -((- - -------) (-5 + Sqrt[5])) + (-----------) (5 + Sqrt[5]) 2 2 2 > ---------------------------------------------------------------}} 10 In[15]:= (* => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3\ > (c+1)] [Joan Z. Yu and Robert Israel in sci.math.symbolic] *) 0. Second In[16]:= RSolve[{r[n] == (1 + c - c^(n-1) - c^(n+1))/(1 - c^n)*r[n - 1] - c*(1 - c^(n-2))/(1 - c^(n-1))*r[n - 2] + 1, r[1] == 1, r[2] == (2 + 2*c + c^2)/(1 + c)}, r[n], n] DSolve::nvld: The description of the equations appears to be ambiguous or invalid. 25.9 Second -2 + n c (1 - c ) r[-2 + n] Out[16]= RSolve[{r[n] == 1 - ------------------------- + -1 + n 1 - c -1 + n 1 + n 2 (1 + c - c - c ) r[-1 + n] 2 + 2 c + c > ------------------------------------, r[1] == 1, r[2] == ------------} n 1 + c 1 - c > , r[n], n] In[17]:= (* Second order ODE with initial conditions---solve first using\ > Laplace transforms: f(t) = sin(2 t)/8 - t cos(2 t)/4 *) 0. Second In[18]:= << Calculus`LaplaceTransform` 2.1 Second In[19]:= ode = f''[t] + 4*f[t] == Sin[2*t] 0. Second Out[19]= 4 f[t] + f''[t] == Sin[2 t] In[20]:= LaplaceTransform[ode, t, s] 0.09 Second Out[20]= -(s f[0]) + 4 LaplaceTransform[f[t], t, s] + 2 2 > s LaplaceTransform[f[t], t, s] - f'[0] == ------ 2 4 + s In[21]:= % /. {f[0] -> 0, f'[0] -> 0} 0. Second 2 Out[21]= 4 LaplaceTransform[f[t], t, s] + s LaplaceTransform[f[t], t, s] == 2 > ------ 2 4 + s In[22]:= Solve[%, LaplaceTransform[f[t], t, s]] 0.02 Second 2 Out[22]= {{LaplaceTransform[f[t], t, s] -> ---------}} 2 2 (4 + s ) In[23]:= Map[InverseLaplaceTransform[#, s, t] &, %[[1, 1]]] 0.04 Second Sin[2 t] t (-Cos[2 t] + --------) 2 t Out[23]= f[t] -> ------------------------ 4 In[24]:= Simplify[%] 0.1 Second -2 t Cos[2 t] + Sin[2 t] Out[24]= f[t] -> ------------------------ 8 In[25]:= (* Now, solve the ODE directly *) 0. Second In[26]:= DSolve[{ode, f[0] == 0, f'[0] == 0}, f[t], t] 0.4 Second Sin[2 t] -(t Cos[2 t]) + -------- 2 Out[26]= {{f[t] -> ------------------------}} 4 In[27]:= Simplify[%] 0.11 Second -2 t Cos[2 t] + Sin[2 t] Out[27]= {{f[t] -> ------------------------}} 8 In[28]:= (* Separable equation => y(x)^2 = 2 log(x + 1) + (4 x + 3)/(x + 1)^2\ > + 2 A *) 0. Second In[29]:= D[y[x], x] == x^2/(y[x]*(1 + x)^3) 0. Second 2 x Out[29]= y'[x] == ------------- 3 (1 + x) y[x] In[30]:= DSolve[%, y[x], x] 0.47 Second Out[30]= {{y[x] -> 2 3 4 x 2 Log[1 + x] > -Sqrt[((1 + x) (-------- + -------- + 2 C[1] + ------------ + 2 2 2 (1 + x) (1 + x) (1 + x) 2 4 x Log[1 + x] 2 x Log[1 + x] 2 > -------------- + ---------------)) / (1 + 2 x + x )]}, 2 2 (1 + x) (1 + x) 2 3 4 x 2 Log[1 + x] > {y[x] -> Sqrt[((1 + x) (-------- + -------- + 2 C[1] + ------------ + 2 2 2 (1 + x) (1 + x) (1 + x) 2 4 x Log[1 + x] 2 x Log[1 + x] 2 > -------------- + ---------------)) / (1 + 2 x + x )]}} 2 2 (1 + x) (1 + x) In[31]:= FullSimplify[%] 2.64 Second 2 2 3 + 4 x + 2 (1 + x) C[1] + 2 (1 + x) Log[1 + x] Out[31]= {{y[x] -> -Sqrt[-------------------------------------------------]}, 2 (1 + x) 2 2 3 + 4 x + 2 (1 + x) C[1] + 2 (1 + x) Log[1 + x] > {y[x] -> Sqrt[-------------------------------------------------]}} 2 (1 + x) In[32]:= (* Homogeneous equation. See Emilio O. Roxin, _Ordinary Differential Equations_, Wadsworth Publishing Company, 1972, p. 11 => y(x)^2 = 2 x^2 log|A x| *) 0. Second In[33]:= D[y[x], x] == y[x]/x + x/y[x] 0. Second x y[x] Out[33]= y'[x] == ---- + ---- y[x] x In[34]:= DSolve[%, y[x], x] 0.27 Second 2 2 Out[34]= {{y[x] -> -(Sqrt[2] Sqrt[x C[1] + x Log[x]])}, 2 2 > {y[x] -> Sqrt[2] Sqrt[x C[1] + x Log[x]]}} In[35]:= Simplify[%] 0.07 Second 2 Out[35]= {{y[x] -> -(Sqrt[2] Sqrt[x (C[1] + Log[x])])}, 2 > {y[x] -> Sqrt[2] Sqrt[x (C[1] + Log[x])]}} In[36]:= (* First order linear ODE: y(x) = [A - cos(x)]/x^3 *) 0. Second In[37]:= x^2*y'[x] + 3*x*y[x] == Sin[x]/x 0. Second 2 Sin[x] Out[37]= 3 x y[x] + x y'[x] == ------ x In[38]:= DSolve[%, y[x], x] 0.12 Second C[1] Cos[x] Out[38]= {{y[x] -> ---- - ------}} 3 3 x x In[39]:= (* Exact equation => x + x^2 sin y(x) + y(x) = A [Roxin, p. 15] *) 0. Second In[40]:= D[y[x], x] == -(1 + 2*x*Sin[y[x]])/(1 + x^2*Cos[y[x]]) 0. Second 1 + 2 x Sin[y[x]] Out[40]= y'[x] == -(-----------------) 2 1 + x Cos[y[x]] In[41]:= DSolve[%, y[x], x] time: command terminated abnormally. real 4143.28 user 4006.17 sys 1.31 ------------------------------------------------------------------------------- Tue Mar 17 16:55:50 MST 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Ordinary Difference and Differential Equations\ > ---------- *) 0. Second In[6]:= (* Second order linear recurrence equationn: r(n) = (n - 1)^2 + m n *) 0. Second In[7]:= << Calculus`LaplaceTransform` 2.09 Second In[8]:= (* Nonlinear ODE => y(x)^3/6 + A y(x) = x + B *) 0. Second In[9]:= eqn = y''[x] + y[x]*y'[x]^3 == 0 0. Second 3 Out[9]= y[x] y'[x] + y''[x] == 0 In[10]:= Simplify[DSolve[%, y[x], x]] 2.66 Second Out[10]= {{y[x] -> 2 C[1] > -------------------------------------------------- + 3 2 1/3 (3 x + Sqrt[-8 C[1] + 9 (x - C[2]) ] - 3 C[2]) 3 2 1/3 > (3 x + Sqrt[-8 C[1] + 9 (x - C[2]) ] - 3 C[2]) }, (1 + I Sqrt[3]) C[1] > {y[x] -> -(--------------------------------------------------) + 3 2 1/3 (3 x + Sqrt[-8 C[1] + 9 (x - C[2]) ] - 3 C[2]) I 3 2 1/3 > - (I + Sqrt[3]) (3 x + Sqrt[-8 C[1] + 9 (x - C[2]) ] - 3 C[2]) }, 2 I (I + Sqrt[3]) C[1] > {y[x] -> -------------------------------------------------- - 3 2 1/3 (3 x + Sqrt[-8 C[1] + 9 (x - C[2]) ] - 3 C[2]) I 3 2 1/3 > - (-I + Sqrt[3]) (3 x + Sqrt[-8 C[1] + 9 (x - C[2]) ] - 3 C[2]) }} 2 In[11]:= (* => y(x) = [3 x + sqrt(1 + 9 x^2)]^(1/3) - 1/[3 x + sqrt(1 + 9\ > x^2)]^(1/3) [Pos96] *) 0. Second In[12]:= Simplify[DSolve[{eqn, y[0] == 0, y'[0] == 2}, y[x], x]] 5.87 Second 2 2/3 -1 + (3 x + Sqrt[1 + 9 x ]) Out[12]= {{y[x] -> ------------------------------}, 2 1/3 (3 x + Sqrt[1 + 9 x ]) 2 2/3 1 + I Sqrt[3] + I (I + Sqrt[3]) (3 x + Sqrt[1 + 9 x ]) > {y[x] -> ---------------------------------------------------------}, 2 1/3 2 (3 x + Sqrt[1 + 9 x ]) 2 2/3 1 - I Sqrt[3] - I (-I + Sqrt[3]) (3 x + Sqrt[1 + 9 x ]) > {y[x] -> ----------------------------------------------------------}} 2 1/3 2 (3 x + Sqrt[1 + 9 x ]) In[13]:= Clear[eqn] 0. Second In[14]:= (* A simple parametric ODE: y(x, a) = A e^(a x) *) 0. Second In[15]:= ode = D[y[x, a], x] == a*y[x, a] 0. Second (1,0) Out[15]= y [x, a] == a y[x, a] In[16]:= DSolve[ode, y[x, a], x] DSolve::nvld: The description of the equations appears to be ambiguous or invalid. DSolve::pde: Partial differential equation may not have a general solution. Try loading Calculus`DSolveIntegrals` to find special solutions. 0.04 Second (1,0) Out[16]= DSolve[y [x, a] == a y[x, a], y[x, a], x] In[17]:= DSolve[ode, y[x, a], {x, a}] 0.16 Second a x Out[17]= {{y[x, a] -> E C[1][a]}} In[18]:= Clear[ode] 0. Second In[19]:= (* ODE with boundary conditions. This problem has nontrivial\ > solutions y(x) = A sin([pi/2 + n pi] x) for n an arbitrary integer *) 0. Second In[20]:= DSolve[{y''[x] + k^2*y[x] == 0, y[0] == 0, y'[1] == 0}, y[x], x] 0.26 Second Out[20]= {{y[x] -> 0}} In[21]:= (* => y(x) = Z_v[sqrt(x)] where Z_v is an arbitrary Bessel function\ > of order v [Gradshteyn and Ryzhik 8.491(9)] *) 0. Second In[22]:= D[y[x], {x, 2}] + 1/x*D[y[x], x] + 1/(4*x)*(1 - v^2/x)*y[x] == 0 0. Second 2 v (1 - --) y[x] x y'[x] Out[22]= ------------- + ----- + y''[x] == 0 4 x x In[23]:= DSolve[%, y[x], x] 0.66 Second Out[23]= {{y[x] -> BesselJ[-v, Sqrt[x]] C[1] + BesselJ[v, Sqrt[x]] C[2]}} In[24]:= (* Delay (or mixed differential-difference) equation. See Daniel\ > Zwillinger, _Handbook of Differential Equations_, Second Edition, Academic Press,\ > Inc., 1992, p. 210 => y(t) = y0 sum((-a)^n (t - n + 1)^n/n!, n = 0..floor(t) +\ > 1) *) 0. Second In[25]:= D[y[t], t] + a*y[t - 1] == 0 0. Second Out[25]= a y[-1 + t] + y'[t] == 0 In[26]:= DSolve[%, y[t], t] DSolve::nvld: The description of the equations appears to be ambiguous or invalid. 0.58 Second Out[26]= {{y[t] -> C[1] - a Integrate[y[-1 + DSolve`t], {DSolve`t, 0, t}]}} In[27]:= (* Discontinuous ODE [Zwillinger, p. 221] => y(t) = cosh t (0 <= t < T) (sin T cosh T + cos T sinh T) sin t + (cos T cosh T - sin T sinh T) cos t (T <= t) *) 0. Second In[28]:= sgn[t_]:= If[t < 0, -1, 1]; 0. Second In[29]:= DSolve[{D[y(t), {t, 2}] + sgn[t - TT]*y[t] == 0, y[0] == 1, y'[0] ==\ > 0}, y[t], t] DSolve::nvld: The description of the equations appears to be ambiguous or invalid. General::stop: Further output of DSolve::nvld will be suppressed during this calculation. 0.07 Second Out[29]= {{y[t] -> 0}} In[30]:= Clear[sgn] 0. Second In[31]:= DSolve[{D[y(t), {t, 2}] + Sign[t - TT]*y[t] == 0, y[0] == 1, y'[0]\ > == 0}, y[t], t] 0.13 Second Out[31]= {{y[t] -> 0}} In[32]:= (* Integro-differential equation. See A. E. Fitzgerald, David E.\ > Higginbotham and Arvin Grabel, _Basic Electrical Engineering_, Fourth Edition, McGraw-Hill Book Company, 1975, p. 117. => i(t) = 5/13 [-8 e^(-4 t) + e^(-t) (8 cos 2 t + sin 2 t)] *) 0. Second In[33]:= eqn = D[i[t], t] + 2*i[t] + 5*Integrate[i[tau], {tau, 0, t}] ==\ > 10*E^(-4*t) 0.03 Second 10 Out[33]= 2 i[t] + 5 Integrate[i[tau], {tau, 0, t}] + i'[t] == ---- 4 t E In[34]:= DSolve[{eqn, i'[0] == 10}, i[t], t] 0.74 Second Out[34]= {{i[t] -> 4 DSolve`t -2 + E Integrate[i[tau], {tau, 0, DSolve`t}] > (-5 Integrate[------------------------------------------------------, 2 DSolve`t E 2 t > {DSolve`t, 0, t}]) / E }} In[35]:= LaplaceTransform[eqn, t, s] 0.28 Second Out[35]= -i[0] + 2 LaplaceTransform[i[t], t, s] + 5 LaplaceTransform[i[t], t, s] 10 > ------------------------------ + s LaplaceTransform[i[t], t, s] == ----- s 4 + s In[36]:= % /. {i[0] -> 0, i'[0] -> 10} 0. Second 5 LaplaceTransform[i[t], t, s] Out[36]= 2 LaplaceTransform[i[t], t, s] + ------------------------------ + s 10 > s LaplaceTransform[i[t], t, s] == ----- 4 + s In[37]:= Solve[%, LaplaceTransform[i[t], t, s]] 0.02 Second 10 s Out[37]= {{LaplaceTransform[i[t], t, s] -> ----------------------}} 2 (4 + s) (5 + 2 s + s ) In[38]:= Map[InverseLaplaceTransform[#, s, t] &, %[[1, 1]]] 0.34 Second 4 Cos[2 t] Sin[2 t] ---------- + -------- t t -4 E 2 E Out[38]= i[t] -> 10 (------- + ---------------------) 4 t 13 13 E In[39]:= Clear[eqn] 0. Second In[40]:= (* System of two linear, constant coefficient ODEs: x(t) = e^t [A cos(t) - B sin(t)], y(t) = e^t [A sin(t) + B cos(t)] *) 0. Second In[41]:= system = {x'[t] == x[t] - y[t], y'[t] == x[t] + y[t]} 0. Second Out[41]= {x'[t] == x[t] - y[t], y'[t] == x[t] + y[t]} In[42]:= DSolve[system, {x[t], y[t]}, t] 0.37 Second Out[42]= {{x[t] -> -I I Sin[(1 + I) t] > C[2] (-- Cos[(1 + I) t] + - Cosh[(1 + I) t] - -------------- + 2 2 2 I > - Sinh[(1 + I) t]) + 2 Cos[(1 + I) t] Cosh[(1 + I) t] I > C[1] (-------------- + --------------- - - Sin[(1 + I) t] + 2 2 2 Sinh[(1 + I) t] > ---------------), y[t] -> 2 I I Sin[(1 + I) t] > C[1] (- Cos[(1 + I) t] - - Cosh[(1 + I) t] + -------------- - 2 2 2 I > - Sinh[(1 + I) t]) + 2 Cos[(1 + I) t] Cosh[(1 + I) t] I > C[2] (-------------- + --------------- - - Sin[(1 + I) t] + 2 2 2 Sinh[(1 + I) t] > ---------------)}} 2 In[43]:= FullSimplify[%] 2.47 Second t Out[43]= {{x[t] -> E (C[1] Cos[t] - C[2] Sin[t]), t > y[t] -> E (C[2] Cos[t] + C[1] Sin[t])}} In[44]:= ( (* Check the answer *) ExpandAll[system /. {x -> Apply[Function, {t, %[[1, 1, 2]]}], y -> Apply[Function, {t, %[[1, 2, 2]]}]}] ) 0.09 Second Out[44]= {True, True} In[45]:= FullSimplify[DSolve[system, {x, y}, t]] 0.26 Second Out[45]= {{x -> -I I Sin[(1 + I) #1] > (C[2] (-- Cos[(1 + I) #1] + - Cosh[(1 + I) #1] - --------------- + 2 2 2 I > - Sinh[(1 + I) #1]) + 2 Cos[(1 + I) #1] Cosh[(1 + I) #1] I > C[1] (--------------- + ---------------- - - Sin[(1 + I) #1] + 2 2 2 Sinh[(1 + I) #1] > ----------------) & ), 2 I I Sin[(1 + I) #1] > y -> (C[1] (- Cos[(1 + I) #1] - - Cosh[(1 + I) #1] + --------------- - 2 2 2 I > - Sinh[(1 + I) #1]) + 2 Cos[(1 + I) #1] Cosh[(1 + I) #1] I > C[2] (--------------- + ---------------- - - Sin[(1 + I) #1] + 2 2 2 Sinh[(1 + I) #1] > ----------------) & )}} 2 In[46]:= ( (* Check the answer *) ExpandAll[system /. First[%]] ) 0.33 Second Out[46]= {True, True} In[47]:= (* Triangular system of two ODEs: x(t) = A e^t [sin(t) + 2], y(t) = A e^t [5 - cos(t) + 2 sin(t)]/5 + B e^(-t) See Nicolas Robidoux, ``Does Axiom Solve Systems of O.D.E.'s Like Mathematica?'', LA-UR-93-2235, Los Alamos National Laboratory, Los Alamos, New Mexico. *) 0. Second In[48]:= system = {x'[t] == x[t] * (1 + Cos[t]/(2 + Sin[t])), y'[t] == x[t] - y[t]} 0. Second Cos[t] Out[48]= {x'[t] == (1 + ----------) x[t], y'[t] == x[t] - y[t]} 2 + Sin[t] In[49]:= DSolve[system, {x[t], y[t]}, t] 1.39 Second t + Log[2 + Sin[t]] Out[49]= {{x[t] -> E C[2], t C[1] E C[2] (-5 + Cos[t] - 2 Sin[t]) > y[t] -> ---- - --------------------------------}} t 5 E In[50]:= Simplify[%] 0.51 Second t Out[50]= {{x[t] -> E C[2] (2 + Sin[t]), t C[1] E C[2] (-5 + Cos[t] - 2 Sin[t]) > y[t] -> ---- - --------------------------------}} t 5 E In[51]:= (* Try solving this system one equation at a time *) 0. Second In[52]:= DSolve[system[[1]], x[t], t] 0.29 Second t + Log[2 + Sin[t]] Out[52]= {{x[t] -> E C[1]}} In[53]:= Simplify[%] 0.07 Second t Out[53]= {{x[t] -> E C[1] (2 + Sin[t])}} In[54]:= DSolve[system[[2]] /. %, y[t], t, DSolveConstants -> (Module[{C}, C]\ > &)] 0.2 Second t C$404[1] E C[1] (-5 + Cos[t] - 2 Sin[t]) Out[54]= {{y[t] -> -------- - --------------------------------}} t 5 E In[55]:= (* 3 x 3 linear system with constant coefficients: (1) real distinct characteristic roots (= 2, 1, 3) [Roxin, p. 109] => x(t) = A e^(2 t), y(t) = B e^t + C e^(3 t), z(t) = -A e^(2 t) - C e^(3 t) *) 0. Second In[56]:= system = {D[x[t], t] == 2*x[t], D[y[t], t] == -2*x[t] + y[t] - 2*z[t], D[z[t], t] == x[t] + 3*z[t]} 0. Second Out[56]= {x'[t] == 2 x[t], y'[t] == -2 x[t] + y[t] - 2 z[t], > z'[t] == x[t] + 3 z[t]} In[57]:= DSolve[system, {x[t], y[t], z[t]}, t] 0.37 Second 2 t t 3 t Out[57]= {{x[t] -> -(E C[2]), y[t] -> E C[1] - E C[3], 2 t 3 t > z[t] -> E C[2] + E C[3]}} In[58]:= (* (2) complex characteristic roots (= 0, -1 +- sqrt(2) i) [Roxin,\ > p. 111] => x(t) = A + e^(-t)/3 [-(B + sqrt(2) C) cos(sqrt(2) t) + (sqrt(2) B - C) sin(sqrt(2) t)], y(t) = e^(-t) [B cos(sqrt(2) t) + C sin(sqrt(2) t)], z(t) = e^(-t) [(-B + sqrt(2) C) cos(sqrt(2) t) -(sqrt(2) B + C) sin(sqrt(2) t)] *) 0. Second In[59]:= system = {D[x[t], t] == y[t], D[y[t], t] == z[t], D[z[t], t] == -3*y[t] - 2*z[t]} 0. Second Out[59]= {x'[t] == y[t], y'[t] == z[t], z'[t] == -3 y[t] - 2 z[t]} In[60]:= DSolve[system, {x[t], y[t], z[t]}, t] 0.28 Second Out[60]= {{x[t] -> -I (-1 - I Sqrt[2]) t -- E (-1 - I Sqrt[2]) t 3 Sqrt[2] E > C[1] + (---------------------- - ---------------------------) C[2] + -I + Sqrt[2] 3 (-I + Sqrt[2]) I (-1 + I Sqrt[2]) t - E (-1 + I Sqrt[2]) t 3 Sqrt[2] E > (--------------------- - ---------------------------) C[3], I + Sqrt[2] 3 (I + Sqrt[2]) (-1 - I Sqrt[2]) t -E I (-1 - I Sqrt[2]) t > y[t] -> (-------------------- + - Sqrt[2] E ) C[2] + 3 3 (-1 + I Sqrt[2]) t -E I (-1 + I Sqrt[2]) t > (-------------------- - - Sqrt[2] E ) C[3], 3 3 (-1 - I Sqrt[2]) t (-1 + I Sqrt[2]) t > z[t] -> E C[2] + E C[3]}} In[61]:= Simplify[%] 3.35 Second Out[61]= {{x[t] -> (-1 - I Sqrt[2]) t t + I Sqrt[2] t > (E (9 E C[1] + 2 I Sqrt[2] t > (-1 - 2 I Sqrt[2]) C[2] + I (I + 2 Sqrt[2]) E C[3]))\ > / 9, y[t] -> I (-1 - I Sqrt[2]) t > - E ((I + Sqrt[2]) C[2] - 3 2 I Sqrt[2] t > (-I + Sqrt[2]) E C[3]), (-1 - I Sqrt[2]) t 2 I Sqrt[2] t > z[t] -> E (C[2] + E C[3])}} In[62]:= (* (3) multiple characteristic roots (= 2, 2, 2) [Roxin, p. 113] => x(t) = e^(2 t) [A + C (1 + t)], y(t) = B e^(2 t), z(t) = e^(2 t) [A + C t] *) 0. Second In[63]:= system = {D[x[t], t] == 3*x[t] - z[t], D[y[t], t] == 2*y[t], D[z[t], t] == x[t] + z[t]} 0. Second Out[63]= {x'[t] == 3 x[t] - z[t], y'[t] == 2 y[t], z'[t] == x[t] + z[t]} In[64]:= DSolve[system, {x[t], y[t], z[t]}, t] 0.25 Second 2 t 2 t 2 t 2 t Out[64]= {{x[t] -> (E + E t) C[1] - E t C[3], y[t] -> E C[2], 2 t 2 t 2 t > z[t] -> E t C[1] + (E - E t) C[3]}} In[65]:= (* x(t) = x0 + [4 sin(w t)/w - 3 t] x0' [Rick Niles] + 6 [w t - sin(w t)] y0 + 2/w [1 - cos(w t)] y0', y(t) = -2/w [1 - cos(w t)] x0' + [4 - 3 cos(w t)] y0 + sin(w t)/w y0' *) 0. Second In[66]:= system = {D[x[t], {t, 2}] == 2*w*D[y[t], t], D[y[t], {t, 2}] == -2*w*D[x[t], t] + 3*w^2*y[t]} 0. Second 2 Out[66]= {x''[t] == 2 w y'[t], y''[t] == 3 w y[t] - 2 w x'[t]} In[67]:= DSolve[system, {x[t], y[t]}, t] 0.82 Second Out[67]= {{x[t] -> 2 2 Cos[t w] > C[1] + C[4] (- - ----------) + C[2] (6 t w - 6 Sin[t w]) + w w 4 Sin[t w] > C[3] (-3 t + ----------), w -2 2 Cos[t w] C[4] Sin[t w] > y[t] -> C[2] (4 - 3 Cos[t w]) + C[3] (-- + ----------) + -------------}} w w w In[68]:= Clear[system] 0. Second In[69]:= (* ---------- Quit ---------- *) 0. Second In[70]:= Quit[] real 27.75 user 25.91 sys 0.63