Nonlinear Differential equation

[spacetime]

Variable co-effiecients Linear Differential equation

$$\frac{d^2 y}{dx^2} = c_1y(1-c_2x)$$

any help? Is there a solution besides a series solution?

 

[HallsofIvy]

First, let me point out that that is NOT a non-linear equation! I wondered about that before the "tex" came up since a "series solution" will not work for a non-linear equation.

It is rather, a "linear equation with variable coefficients". I don't see any method other than a series solution which should work nicely.

 

[dextercioby]

For future reference,an ODE is said to be NONLINEAR in three possble cases
1.The power of the derivative(s) is not "1".
2.The power of the unknown function is not "1".
3.Cases 1 & 2 at the same time...

Daniel.

 

[vincentchan]

try substitude u = 1 - cx,

 

[arildno]

The solutions are the Airy family of functions in disguise:
Let:
$$u=ax+b$$
where "a,b" are constants to be determined.
Then:
$$\frac{d^{2}y}{dx^{2}}=a^{2}\frac{d^{2}y}{du^{2}}$$
In order to determine "a,b", we require:
$$\frac{c_{1}-c_{1}c_{2}x}{a^{2}}=ax+b=u$$
This yields:
$$a=-(c_{1}c_{2})^{\frac{1}{3}},b=(\frac{c_{1}}{c_{2}^{2}})^{\frac{1}{3}}$$
And with these choices:
$$\frac{d^{2}y}{du^{2}}=uy$$
This is the Airy differential equation.
The power series solutions(i.e, Airy functions) are well studied.

 

[saltydog]

For future reference,an ODE is said to be NONLINEAR in three possble cases
1.The power of the derivative(s) is not "1".
2.The power of the unknown function is not "1".
3.Cases 1 & 2 at the same time...

Daniel.

It' also considered non-linear if the dependent variable is contained in a transcendental function; the non-linear pendulum being the canonical example:


$$\frac{d^2\theta}{d t^2} + (g/L)\sin{\theta} = 0$$

You know, when you have a pendulum on a rigid rod and push it so hard it goes round and round.