Solutions of $\varphi'' = \alpha + \beta \varphi + \gamma \varphi^2$

Question

Is there a general method to find the solutions of the equation $$\varphi'' = \alpha + \beta \varphi + \gamma \varphi^2$$

And more generally, is there a method to find solutions of a polynomial differential equation $\mathrm P(\varphi^{(n)}, ... , \varphi) = 0$ ?

Answer 1

If $\alpha,\beta,\gamma$ are constants, and $\gamma\neq 0$ then you can use

$\phi = \varphi + \dfrac{\beta}{2\gamma}$ to make problem as

$\phi''= A+B\phi^2$

Then multiply $\phi'$ on both sides

$2\phi'\phi'' = (\phi'^2)' = 2A\phi'+\dfrac{2}{3}B(\phi^3)' = 2A+2B\phi^2\phi'$

then integrate,

$\phi' =\sqrt{ 2A\phi + \dfrac{2}{3}B(\phi^3) + C}$

When

$A = 0$, it is not hard to see that

$$\int \dfrac{d\phi}{\sqrt{\dfrac{2}{3}B\phi^3+C}} = x+C_1$$

I think there is some formula about this.

When

$A\neq 0$, then you probably will see

$$\phi = \int\sqrt{2A\phi + \dfrac{2}{3}B\phi^3+C}dx$$

if we take the operator $T:C^1(I)\rightarrow C^1(I)$ as the integral operator, $I$ is some small interval, under some situation, we can assume that this operator can give you a contraction map.

Then $\phi = \lim_{n\rightarrow\infty}T^n\phi_0$.