:

## A first-order linear ODE with one general (possibly small) non-linear term : can it be solved?

Hi everyone,

I am looking at the following ODE:

x'(t) + a(t)*x(t)  - t*a(t) + b*a(t)*f(x(t)) = 0

where x is the function of t to be found
with t>=0, x(t)>0
with initial condition x(0)=x0 > 0 given
with a(t) a known function of t inside the positive quadrant, a(t)>0
with f(x) a known function of x inside the positive quadrant, f(x)>0
with b a known positive real constant, b>0

If b=0, we have a first-order linear ODE, but I'm interested in b>0.
If f(x)=x, we have again a first-order linear ODE
If f(x)=x^2, we have a Riccati equation, that can be transformed to a second-order linear ODE, and, if I remember correctly, that can be solved if I know a particular solution.
If f(x)=x^r, for some real number r, we have a non-homogeneous first-order ODE of Bernoulli type, and I cannot remember if and how it can be solved

However, I am interested in solving the equation in its full generality, without specific assumptions about the function f(x). That may not be possible. Is it?

I would be interested in any of the following:

1. A theorem that would state conditions on f(x) such that the equation may be solved, together with the solution.

2. An impossibility theorem that would state that under the conditions a(t)>0, b>0, the equation cannot be solved for a general function f(x)

3. An approximation theorem for small b>0. I'm actually very interested in the case where b is small, for in my problem it is small, and I'm also interested in the limit case b --> 0.

The following did not help, but I bet I am not using Maple's full capabilities on this one!

restart: with(DEtools):
ode:= diff(x(t),t) + a(t)*x(t)  - t*a(t) + b*a(t)*f(x(t)) = 0;
infolevel[dsolve] := 4: 