# Thread: strange differntial equation

1. ## strange differntial equation

Hi all,

can someone tell me what the general solution is for the following differential equation.

z''+z=b*z^3 (z^3 meaning "z" to the third power, not the third derivative of z, while z'' stands for the second derivative of z)

Is it possible that there is no solution for it.

How do you in heavens name solve such differential equations?

2. Dear polo,

This differential equation could be solved in several ways. But I will give you the most easiest I can think of. In a differential equation if x is absent explicitily, (does not have $x^{1}$ terms.) this method can be used.

$\frac{d^2z}{dx^2}+z=bz^{3}$

Substitute, $p=\frac{dz}{dx}$

$\frac{dp}{dx}=\frac{dp}{dz}\times\frac{dz}{dx}=p\f rac{dp}{dz}$

$\frac{dp}{dx}+z=bz^{3}$

$p\frac{dp}{dz}+z=bz^{3}$

Seperation of variables and integrating yields, $\frac{p^2}{2}=\int{(bz^{3}-z)dz}$

$\frac{p^2}{2}=\frac{bz^{4}}{4}-\frac{z^{2}}{2}+C~;Where~C~is~an~arbitary~constant .$

3. Originally Posted by Sudharaka
Dear polo,

This differential equation could be solved in several ways. But I will give you the most easiest I can think of. In a differential equation if x is absent explicitily, (does not have $x^{1}$ terms.) this method can be used.

$\frac{d^2z}{dx^2}+z=bz^{3}$

Substitute, $p=\frac{dz}{dx}$

$\frac{dp}{dx}=\frac{dp}{dz}\times\frac{dz}{dx}=p\f rac{dp}{dz}$

$\frac{dp}{dx}+z=bz^{3}$

$p\frac{dp}{dz}+z=bz^{3}$

Seperation of variables and integrating yields, $\frac{p^2}{2}=\int{(bz^{3}-z)dz}$

$\frac{p^2}{2}=\frac{bz^{4}}{4}-\frac{z^{2}}{2}+C~;Where~C~is~an~arbitary~constant .$

Thanks Sudharaka,

I seems to me that there is no particular solution to this differential equation; intuitively it looks to me that we are dealing with a helix since the differential equation z''+z=b is an ellipse and obeys Keplers first law
My mathematical skills need some revision (it has been a long time)

4. Originally Posted by polo
Thanks Sudharaka,

I seems to me that there is no particular solution to this differential equation; intuitively it looks to me that we are dealing with a helix since the differential equation z''+z=b is an ellipse and obeys Keplers first law
My mathematical skills need some revision (it has been a long time)

Sorry a spiral i mean...it looks like a spiral

5. Originally Posted by polo
Sorry a spiral i mean...it looks like a spiral
Hi Sudharaka,

Maybe I should be little bit more clear on my question
I have include a word document where I have outlined the problem a little bit better.

I have done it in word because for the moment I don't know how to write mathematical formulas in this text box.

6. Dear Polo,

In the above solution I have used you will have you substitute for p and integrate it another time to get the final answer, which is out of my scope. But if you use the Wolfram integrator you could find the solution as,

http://integrals.wolfram.com/index.j...D&random=false or http://integrals.wolfram.com/index.j...D&random=false

The answers contains imaginary numbers. So I do not think it will be a circle or an ellipse.