ode_iota_omega

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Dismod ODE With Constant iota and omega

If \(iota\) and \(\omega\) are constant w.r.t. age and time, and the other rates are zero, the dismod ode is

\[\begin{split}S'(a) &= - (\omega + \iota) S(a) \\ C'(a) &= + \iota S(a) - \omega C(a)\end{split}\]

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

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

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

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

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

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

Dividing both sides of the last equation by \(D(a)\) we obtain

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

As a check of this solutions we compute

\[\begin{split}C'(a) & = (\omega + \iota) \exp[ - (\omega + \iota) a ] - \omega \exp( - \omega a ) \\ C'(a) + \omega C(a) & = \iota \exp[ - (\omega + \iota) a ] \\ C'(a) + \omega C(a) &= \iota S(a)\end{split}\]

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