Analytic Solution of Second Order Nonlinear odes Any idea how to find analytic solution of the following ODE.
$y''+0.1 y'+y^{5} = \sin (t)$
I will really appreciate your response!
Shah
 A: I can suggest you 2 ways of solving this equation.


*

*Since the ODE is non-homogeneous, I would suggest you to apply the nonlinear Green's function method described here: https://arxiv.org/pdf/1806.00274.pdf. Even though your case is not considered here, you can do similar calculations.


Let me shortly describe the solution. So, you have the ODE
$$
y'' + 0.1 y' + y^5 = \sin t.
$$
Its solution admits the short time expansion
$$
y(t) = \int_0^t G\left(\tau\right) \sin(t - \tau) d\tau + \sum_{k = 1}^\infty y_k \int_0^t \tau^k G\left(\tau\right) \sin(t - \tau) d\tau,
$$
where $a_k$ are determined in terms of $y^k(0)$, $G$ is the so-called nonlinear Green's function. As it is proved in https://arxiv.org/pdf/1805.10495.pdf,
$$
G(t) = \theta(t) y_0(t),
$$
where $\theta$ is the Heaviside function and $y_0$ is the solution of the homogenous equation
$$
y'' + 0.1 y' + y^5 = 0,
$$
subjected to the non-homogeneous Cauchy conditions:
$$
y(0) = 0, y'(0) = 1.
$$
Neither wolfram mathematica, nor maple 17 gives $y_0$ explicitly, but you can try a power series solution.


*Just use Adomian decomposition method. It's relatively easier to apply in your case.

