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If I'm given the homogenous first-order linear ODE in one dimension

$\frac{dx}{dt} = ax(t), \ \ x_0 = 1$,

a straightforward calculation shows that

$x(t) = e^{a t}$.

Now if I add some driving function $f$ that depends only on the current time $t$, the dynamics becomes

$\frac{dx}{dt} = ax(t) + f(t), \ \ x_0 = 1$.

The solution to the inhomogenous ODE is

$x(t) = e^{a t} + \int_0^t e^{a (t-u)}f(u)du$,

so we can relate the solution of the inhomogenous equation to that of the homogenous equation and a certain operation on the driving function (in this case, a kind of exponential damping or amplification).

I'm wondering if there are any results that generalise this to the nonlinear setting (still first-order). If I know the solution $x(t)$ of the homogenous equation, is there some result on the relationship between $x$ and the solution to the ODE once I include a driving function?

If no general result exists, are there conditions that I can impose on the dynamics to deal with this kind of thing in special cases?

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1 Answer 1

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What you are doing is basically a pertubation expansion. In the case of a linear ode we are lucky, that any superposition of solutions is still a solution (easy calculation, left as an exercise). For nonlinear odes this is not the case.

Still, the homogenous solution is not useless. Consider an examplary equation $$ x' + A(t)x^2 = \epsilon f(t) $$ where I have added the $\epsilon$ as a pertubation parameter, that we want to set to 1 at the end of our calculation. The first term in this pertubation expansion will be the solution of the homogenious ode $x'+A(t)x^2=0$. After that we have to consider an (most likely) infinite number of correction terms though, to get the (asymptotically) correct answer.

(Whether this is the best place to add the parameter $\epsilon$ depends largly on the $A(t)$ and $f(t)$.)

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