Mathematics Asked by BenG73 on February 19, 2021

How to find the solution $y(t)$ of the following ODE provided that:

- $y(t)$ has a strictly positive value at time $t=0$
- $mu(t)$, $alpha(t)$ and $sigma(t)$ are known functions of time with all "suitable" properties.
- $y(t)$ has a strictly positive value at time $t=0$?

$$ partial_{t}y(t)=sigma(t)-alpha(t) yleft(tright)-muleft(tright)y(t)^{2}$$

May be finding a solution for a more simple problem may help. For example, i study the case where both $sigma(t)$ and $alpha(t)$ are constant but not $mu(t)$

Without the term $sigma(t)$, it looks like a Bernoulli ODE but how to deal with this additional term?

$$ frac{partial y}{partial t}=sigma(t)-alpha(t) yleft(tright)-muleft(tright)y(t)^{2} tag 1$$ It is common to solve a Riccati ODE if a particular solution is known. In the present case no particular solution can be found by inspection. So, in this case it is of interest to transform the non-linear first order ODE into a linear second order ODE because a linear ODE is generaly easier to solve than a non-linear ODE even if the transformation increases the order of the ODE.

The usual method involves the change of function : $$y(t)=frac{u'(t)}{mu(t)u(t)}$$

$$y'=-frac{u''}{mu u}-frac{mu'u'}{mu^2u}-frac{u'^2}{mu u^2}=sigma-alphafrac{u'}{mu u}-muleft(frac{u'}{mu u} right)^2$$ After simplification : $$-frac{u''}{mu u}-frac{mu'u'}{mu^2u}=sigma-alphafrac{u'}{mu u}$$ $$u''+(frac{mu'}{mu}-alpha)u'+sigmamu :u=0$$ Let $quad frac{mu'(t)}{mu(t)}-alpha(t)=f(t)quad$ and $quadsigma(t)mu(t)=g(t)$ $$u''(t)+f(t)u'(t)+g(t)u(t)=0$$ This is the general second order linear ODE. There is no standard special function available to express the solutions in the general case. Only for a few kind of functions $f(t)$ and $g(t)$ the solution can be explicitly expressed thanks to a limited number of elementary and available special functions.

Thus don't expect an explicit analytical solution of your equation $(1)$ for any functions $sigma(t):,:alpha(t):,:mu(t)$ .

Even in the case $sigma=$any constant and $alpha=$any constant and $mu(t)=$ any not constant function, the functions $f(t)$ and $g(t)$ remain any related functions. The conclusion remains the same : Don't expect a explicit solution with a finite number of standard and special functions, except for a few kind of functions $mu(t)$.

The function $mu(t)$ must be specified in the wording of the question if you want to know if the solution of Eq.$(1)$ can be or not explicitly expressed.

For example if $mu(t)=frac{c_1}{t+c_2}$ and $sigma$ , $alpha=$ constants the solution $y(t)$ involves the "confluent hypergeometric" functions.

Answered by JJacquelin on February 19, 2021

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