Analytical solution of a nonlinear ordinary differential equation

Question

I would like to find the exact solution of the following nonlinear ODE.

$$ A_1\frac{d^2 \rho}{dx^2}+(1-2 \rho + A_2)\frac{d \rho}{dx}-A_3\rho+A4=0, \hspace{10mm} (eq.1)$$
with $$\rho(0)=b_1 , \hspace{10mm} \rho(1)=b_2.$$

Here $\rho$ is a differentiable function and $A_1, A_2, A_3, A_4, b_1$ and $b_2$ are real constants.

My attempt: I tried to convert this equation into first-order equation by substituting

$$w(\rho)=\frac{d\rho}{dx}$$ which leads to $$A_1 w \frac{dw}{dp}=(2\rho-A2-1)w+A3\rho-A4.$$

Letting $2\rho+c=z$ where $c=-A_2-1$ gives

$$2A_1w\frac{dw}{dz}=z(w+\frac{A3}{2})-\frac{A3}{2}c-A4.$$

which can be written as $$c_1ww^{'}=z(w+c_2)+c_3 \hspace{10mm} (eq.2)$$
where $c_1=2A_1$, $c_2=\frac{A_3}{2}$, $c_3=-\frac{A_3}{2}-A_4$.

I don't know how to proceed further and solve (eq.2).

Remark: I have solved this equation numerically. So I want an analytical solution for this equation.

Could anyone among you help me in solving eq.(1) or eq.(2) analytically? Any help is appreciated.

Thank you.

Answer 1

Hint:

your last equation in the homogeneous case has the form $$yy'=a(y+b) \Rightarrow \frac{yy'}{y+b}=a \Rightarrow y'(\frac{y+b}{y+b}-\frac{b}{y+b})=a \Rightarrow y'-\frac{by'}{y+b}=a $$ $$ \Rightarrow \int y'-\int \frac{by'}{y+b}= \int a $$

Answer 2

$$c_1w(z)\frac{dw}{dz}=(w(z)+c_2)z+c_3 \hspace{10mm} (eq.2)$$ Let $\quad w(z)=\frac{1}{u(z)}$

$-c_1\frac{1}{u^3}\frac{du}{dz}=(\frac{1}{u(z)}+c_2)z+c_3 $

$$\frac{du}{dz}= -\frac{1}{c_1}(c_2z+c_3)u^3 -\frac{z}{c_1}u^2$$

This is an Abel's Differential Equation of first kind.

These kind of equations are not solvable in general (except particular cases) with elementary and standard special functions.

https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf

Answer 3

Let $w=\dfrac{d\rho}{dx}$ ,

Then $\dfrac{d^2 \rho}{dx^2}=\dfrac{dw}{dx}=\dfrac{dw}{d\rho}\dfrac{d\rho}{dx}=w\dfrac{dw}{d\rho}$

$\therefore A_1w\dfrac{dw}{d\rho}+(1-2\rho+A_2)w-A_3\rho+A_4=0$

$A_1w\dfrac{dw}{d\rho}=(2\rho-A_2-1)w+A_3\rho-A_4$

Let $s=\rho-\dfrac{A_2+1}{2}$ ,

Then $A_1w\dfrac{dw}{ds}=2sw+A_3s+\dfrac{(A_2+1)A_3}{2}-A_4$

Let $t=\dfrac{s^2}{A_1}$ ,

Then $\dfrac{dw}{ds}=\dfrac{dw}{dt}\dfrac{dt}{ds}=\dfrac{2s}{A_1}\dfrac{dw}{dt}$

$\therefore2sw\dfrac{dw}{dt}=2sw+A_3s+\dfrac{(A_2+1)A_3}{2}-A_4$

$w\dfrac{dw}{dt}=w+\dfrac{A_3}{2}+\dfrac{(A_2+1)A_3-2A_4}{4s}$

$w\dfrac{dw}{dt}=w+\dfrac{A_3}{2}\pm\dfrac{(A_2+1)A_3-2A_4}{4\sqrt{A_1t}}$

This belongs to an Abel equation of the second kind in the canonical form.

Please follow the method in https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf or in http://www.iaeng.org/IJAM/issues_v43/issue_3/IJAM_43_3_01.pdf