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Lecture 43, System of ODE with solution eigenvalues

System of ODE with solution eigenvalues and eigenvectors

Given unknown vector X, derivatives of X as X', and known matrix A
one possible system of ordinary differential equations might be:

 X(t)' = A X(t)

Computing the eigenvalues %lambda; and corresponding eigenvectors v
of A, the general solution is of the form:

 X(t) = sum { c_i * exp(%lambda;_i * t) * v_i }

A simple sample system of equations is

                                   known constants an t = same value
 |x(t)'|   | 1  3  5 |   | x(t) |    with x(0) = a1   x'(0) = d1
 |y(t)'| = | 2  6  7 | * | y(t) |         y(0) = a2   y'(0) = d2
 |z(t)'|   | 8  9  4 |   | z(t) |         z(0) = a3   z'(0) = d3

expanding to system of ODE equations with three unknown functions
x(t), y(t), z(t)

  d x(t)/dt = x(t)' = 1*x(t) + 3*y(t) + 5*z(t)
  d y(t)/dt = y(t)' = 2*x(t) + 6*y(t) + 7*z(t)
  d z(t)/dt = z(t)' = 8*x(t) + 9*y(t) + 4*z(t)

having a general solution

find eigen values l1, l2, l3 and eigen vectors v1, v2, v3 for
 | 1  3  5 |
 | 2  6  7 |
 | 8  9  4 |

  x(t) = c1 * exp(l1*t) * v1 + c2 * exp(l2) * v2 + c3 * exp(l3) * v3
  y(t) = c1 * exp(l1*t) * v1 + c2 * exp(l2) * v2 + c3 * exp(l3) * v3
  z(t) = c1 * exp(l1*t) * v1 + c2 * exp(l2) * v2 + c3 * exp(l3) * v3

substituting initial conditions gives 3 equations in 3 unknowns
(simplified at t=0, exp(l1*t) = 1, else compute numeric value

  | x(0) | = | v1 v2 v3 |   | c1 |
  | y(0) | = | v1 v2 v3 | * | c2 |
  | z(0) | = | v1 v2 v3 |   | c3 |

substitute c1, c2, c3 in general solution

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