# Interesting 2nd order non-linear DE

• Feb 2nd 2011, 11:32 AM
KrayzBlu
Interesting 2nd order non-linear DE
Hello,

I have recently come across an interesting second order non-linear homogeneous DE whose form I cannot seem to find a general solution to. It loos like such:

$\displaystyle \dfrac{d^2}{dx^2}f(x) + B(x)*\dfrac{d}{dx}f(x) + f(x)*\dfrac{d}{dx}B(x) = 0$

Where B(x) is a arbitrary function. Has anyone seen this before or have any hints?

Thanks
• Feb 2nd 2011, 11:36 AM
TheEmptySet
Quote:

Originally Posted by KrayzBlu
Hello,

I have recently come across an interesting second order non-linear homogeneous DE whose form I cannot seem to find a general solution to. It loos like such:

$\displaystyle \dfrac{d^2}{dx^2}f(x) + B(x)*\dfrac{d}{dx}f(x) + f(x)*\dfrac{d}{dx}B(x) = 0$

Where B(x) is a arbitrary function. Has anyone seen this before or have any hints?

Thanks

Notice that

$\displaystyle \displaystyle \frac{d}{dx}\left[B(x)f(x)\right]=B(x)\frac{df}{dx}+f(x)\frac{dB}{dx}$

This gives the equation as

$\displaystyle \displaystyle \frac{d}{dx}\left[ \frac{df}{dx}+B(x)f(x)\right]=0$

Now just integrate both sides and you have a first order equation that can be solved via an integrating factor.
• Feb 2nd 2011, 11:41 AM
Ackbeet
It's nonlinear?
• Feb 2nd 2011, 01:15 PM
KrayzBlu
Thanks
You're right! I'm embarrassed that I missed that (Itwasntme).

Thank you, sir.

(Although interestingly, alpha couldn't solve it either f&#39;&#39;&#40;x&#41; &#43; b&#40;x&#41;&#42;f&#39;&#40;x&#41; &#43; b&#39;&#40;x&#41;&#42;f&#40;x&#41; &#61; 0 - Wolfram|Alpha)

As to Ackbeet, from what I understand, and from what alpha seems to confirm, if B(x) could be non-linear, then the equation is non-linear. Please correct me if I'm wrong! (I could never quite get this straight in school)
• Feb 2nd 2011, 01:27 PM
TheEmptySet
Quote:

Originally Posted by KrayzBlu
You're right! I'm embarrassed that I missed that (Itwasntme).

Thank you, sir.

(Although interestingly, alpha couldn't solve it either f&#39;&#39;&#40;x&#41; &#43; b&#40;x&#41;&#42;f&#39;&#40;x&#41; &#43; b&#39;&#40;x&#41;&#42;f&#40;x&#41; &#61; 0 - Wolfram|Alpha)

As to Ackbeet, from what I understand, and from what alpha seems to confirm, if B(x) could be non-linear, then the equation is non-linear. Please correct me if I'm wrong! (I could never quite get this straight in school)

I would not classify it as nonlinear. The equation just has non constant coefficients. It would only be nonlinear if$\displaystyle B(x)$ somehow depened on $\displaystyle f(x)$ as an input.

For example is $\displaystyle B(x)=\sin(x) \implies B'(x)=\cos(x)$

Then the equation would be

$\displaystyle f''(x)+\sin(x)f'(x)+cos(x)f(x)=0$

This is a 2nd order linear ODE