#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User User #3: " ---------- Ordinary Difference and Differential Equations ---------- " User #4: " Second order ODE with initial conditions: f(t) = sin(2 t)/8 - t cos(2 t)/4 " #5: q1_ :== ODE(f__ + 4*f = SIN(2*t), t, f, f_, f__) User Simp(#5) SIN(2*t)*COS(4*t) #6: f = c2_*SIGN(t)*SIN(2*t) - ------------------- + 16 COS(2*t)*SIN(4*t) / t \ SIN(2*t) ------------------- + |c1_ - ---|*COS(2*t) + ---------- 16 \ 4 / 16 #7: q2_ :== APPLY_IC(RHS(q1_), [t, 0], [f, 0], [f1, 0]) User Simp(#7) SIN(2*t)*COS(4*t) COS(2*t)*SIN(4*t) t*COS(2*t) #8: - ------------------- + ------------------- - ------------ + 16 16 4 SIN(2*t) ---------- 16 t*COS(2*t) #9: q2_ + ------------ User 4 Simp(#9) SIN(2*t)*COS(4*t) COS(2*t)*SIN(4*t) SIN(2*t) #10: - ------------------- + ------------------- + ---------- 16 16 16 #11: " Simplify the expression above " User #12: Trigonometry := Collect User SIN(2*t) #13: ---------- Simp(#10) 8 #14: Trigonometry := Auto User User #15: " Separable equation => y(x)^2 = 2 log(x + 1) + (4 x + 3)/(x + 1)^2 + 2 A " / 2 \ | x | #16: ODE|y_ = ------------, x, y, y_| User | 3 | \ y*(1 + x) / 2 4*x + 3 y = 2*LN(x + 1) + ---------- + c1_ #17: 2 Simp(#16) (x + 1) User #18: " Homogeneous equation. See Emilio O. Roxin, _Ordinary Differential " User #19: " Equations_, Wadsworth Publishing Company, 1972, p. 11 " #20: " => y(x)^2 = 2 x^2 log|A x| " User / y x \ #21: ODE|y_ = --- + ---, x, y, y_| User \ x y / 2 2 y /(2*x ) #22: #e Simp(#21) ------------- = c1_ x #23: " First order linear ODE: y(x) = [A - cos(x)]/x^3 " User / 2 SIN(x) \ #24: q_ :== ODE|x *y_ + 3*x*y = --------, x, y, y_| User \ x / 3 #25: - COS(x) - x *y = c1_ Simp(#24) #26: FACTOR(SOLVE(q_, y)) User / COS(x) + c1_ \ |y = - --------------| #27: | 3 | Simp(#26) \ x / User #28: " Exact equation => x + x^2 sin y(x) + y(x) = A [Roxin, p. 15] " / 1 + 2*x*SIN(y) \ ODE|y_ = - ----------------, x, y, y_| #29: | 2 | User \ 1 + x *COS(y) / 2 #30: x *SIN(y) + x + y = c1_ Simp(#29) #31: " Nonlinear ODE => y(x)^3/6 + A y(x) = x + B " User 3 #32: q_ :== ODE(y__ + y*y_ = 0, x, y, y_, y__) User Simp(#32) / 2 \ #33: \false = 0, SQRT(alpha)*SIGN(y) - c2_*SQRT(alpha*y - 2) = 0/ User #34: " => y(x) = [3 x + sqrt(1 + 9 x^2)]^(1/3) - 1/[3 x + sqrt(1 + 9 x^2)]^(1/3) " #35: " [Pos96] " User #36: APPLY_IC(RHS(q_), [x, 0], [y, 0], [y1, 2]) User #37: [y = @2] Simp(#36) User #38: " ODE with boundary conditions. This problem has nontrivial solutions " User #39: " y(x) = A sin([pi/2 + n pi] x) for n an arbitrary integer " 2 #40: q_ :== ODE(y__ + k *y = 0, x, y, y_, y__) User #41: y = c2_*SIGN(x)*SIN(k*x) + c1_*COS(k*x) Simp(#40) #42: APPLY_IC(RHS(q_), [x, 0], [y, 0]) User #43: 0 Simp(#42) User #44: " => y(x) = Z_v[sqrt(x)] where Z_v is an arbitrary Bessel function of order v " #45: " [Gradshteyn and Ryzhik 8.491(9)] " User User / / 2 \ \ | 1 1 | v | | #46: ODE|y__ + ---*y_ + -----*|1 - ----|*y = 0, x, y, y_, y__| \ x 4*x \ x / / #47: y = c1_*JN(|v|, SQRT(x)) + c2_*YN(|v|, SQRT(x)) Simp(#46) #48: " Discontinuous ODE [Zwillinger, p. 221] " User #49: " => y(t) = cosh t (0 <= t < T) " User #50: " (sin T cosh T + cos T sinh T) sin t " User User #51: " + (cos T cosh T - sin T sinh T) cos t (T <= t) " #52: SGN(t) := IF(t < 0, -1, 1) User #53: q_ :== ODE(y__ + SGN(t - t_)*y = 0, t, y, y_, y__) User Simp(#53) / 1 \ #54: y = c1_*SQRT(t)*JN|---, t*SQRT(IF(1 - t_ < 0, -1, 1))| + \ 2 / / 1 \ c2_*SQRT(t)*YN|---, t*SQRT(IF(1 - t_ < 0, -1, 1))| \ 2 / #55: APPLY_IC(RHS(q_), [t, 0], [y, 1], [y1, 0]) User #56: 0 Simp(#55) #57: q_ :== ODE(y__ + SIGN(t - t_)*y = 0, t, y, y_, y__) User Simp(#57) / 1 \ #58: y = c1_*SQRT(t)*JN|---, t*SQRT(- SIGN(t_ - 1))| + \ 2 / / 1 \ c2_*SQRT(t)*YN|---, t*SQRT(- SIGN(t_ - 1))| \ 2 / #59: APPLY_IC(RHS(q_), [t, 0], [y, 1], [y1, 0]) User #60: 0 Simp(#59) User #61: " Triangular system of two ODEs: x(t) = A e^t [sin(t) + 2], " #62: " y(t) = A e^t [5 - cos(t) + 2 sin(t)]/5 + B e^(-t) "User User #63: " See Nicolas Robidoux, ``Does Axiom Solve Systems of O.D.E.'s Like " User #64: " Mathematica?'', LA-UR-93-2235, Los Alamos National Laboratory, Los Alamos, " #65: " New Mexico. " User / / COS(t) \ \ #66: system := |x_ = x*|1 + ------------|, y_ = x - y| User \ \ 2 + SIN(t) / / #67: " Try solving this system one equation at a time " User q1_ :== ODE(system , t, x, x_) #68: 1 User -t x*#e #69: - ------------ = c1_ Simp(#68) SIN(t) + 2 #70: FACTOR(SOLVE(q1_, x)) User / t \ #71: \x = - c1_*#e *(SIN(t) + 2)/ Simp(#70) t #72: x := - c1_*#e *(SIN(t) + 2) User q2_ :== ODE( lim system , t, y, y_) #73: c1_->c2_ 2 User Simp(#73) t 2*t / c2_*COS(t) 2*c2_*SIN(t) \ #74: y*#e - #e *|------------ - -------------- - c2_| = c1_ \ 5 5 / #75: SOLVE(q2_, y) User Simp(#75) / t / c2_*COS(t) 2*c2_*SIN(t) \ -t\ #76: |y = #e *|------------ - -------------- - c2_| + c1_*#e | \ \ 5 5 / / #77: x := User #78: system := User #79: " ---------- Quit ---------- " User