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lines 15-85 of file: xrst/math.xrst
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{xrst_begin ode_iota_omega}
{xrst_spell
  da
  ds
}

Dismod ODE With Constant iota and omega
#######################################
If :math:`iota` and :math:`\omega` are constant w.r.t. age and time,
and the other rates are zero, the dismod ode is

.. math::

        S'(a) &= - (\omega + \iota) S(a) \\
        C'(a) &= + \iota  S(a) - \omega C(a)

Using the first equation with initial condition :math:`S(0) = 1`,
we have

.. math::

    S(a) = \exp[ - (\omega + \iota)  a ]

The integrating factor for the second equation is
:math:`D(a) = \exp( \omega a )` .
The derivative of :math:`D(a) C(a)` is

.. math::

    \frac{d}{da} [ D(a) C(a) ]
        &= D'(a) C(a) + D(a) C'(a) \\
        &= \omega D(a) C(a) + \iota D(a) S(a) - \omega D(a) C(a)  \\
        &= \iota D(a) S(a)

Note that the right hand side does not depend on :math:`C(a)` .
Integrating both sides of the last equation above from zero to :math:`a`,
and using the initial condition :math:`C(0) = 0`, we obtain

.. math::

    D(a) C(a)
        &= \iota \int_0^a D(s) S(s) ds \\
        &= \iota \int_0^a \exp( \omega s ) \exp[ -(\omega + \iota) s ] ds \\
        &= \left[ - \exp( - \iota s ) \right]_0^a \\
        &= 1 - \exp( - \iota a )

Dividing both sides of the last equation by :math:`D(a)` we obtain

.. math::

    C(a) = \exp( - \omega a ) - \exp[ - (\omega + \iota) a ]

As a check of this solutions we compute

.. math::

    C'(a)
        & = (\omega + \iota) \exp[ - (\omega + \iota) a ]
            - \omega \exp( - \omega a )
    \\
    C'(a) + \omega C(a)
        & = \iota \exp[ - (\omega + \iota) a ]
    \\
    C'(a) + \omega C(a)
        &= \iota S(a)

Furthermore :math:`C(0) = 0` . Thus :math:`C(a)` satisfies its
differential equation and its initial condition.


{xrst_end ode_iota_omega}