Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I've just gotten back a corrected homework about differential equations, and now I need your help: Why is the ODE $u''(x)=u(x)\sqrt{x}$ homogeneous, but the PDE $u_{xx}(x,y)+u_{yy}(x,y)e^{\sin x}=1$ is inhomogeneous? In both cases we have a function of $x$ that is not related to $u$, namely $e^{\sin x}$ and $\sqrt{x}$, don't we? So I'd think that both are inhomogeneous.

What am I doing wrong here?

Cheers, Marie :)

share|improve this question

1 Answer 1

up vote 5 down vote accepted

Don't mix up notions of autonomous ODEs (where no direct instance of the independent variable can appear) and linear homogeneous equations. The equation $$ u''(x) - u(x)\sqrt x = 0 $$ is homogeneous since the RHS is zero but not autonomous due to the term $\sqrt{x}.$ W.r.t. the PDE $$ u_{xx}+u_{yy} \mathrm e^{\sin x} = 1 $$ the RHS is non-zero, so the PDE is not homogeneous.

Some more examples:

  1. homogeneous autonomous $$ u'(x)+u(x) = 0. $$

  2. homogeneous non-autonomous $$ u''(x)+\color{red}{x}\cdot u(x) = 0 $$

  3. non-homogeneous autonomous $$ u'(x)-2u(x) = \color{red}{1} $$

  4. non-homogeneous non-autonomous $$ u''(x)+\color{red}{x}\cdot u'(x) = \color{red}{x^2+1} $$

where red color is used to highlight terms which bring "non" into the classification.

share|improve this answer
    
okay, the RHS is supposed to be $=1$, sorry! –  Marie. P. Mar 22 '12 at 9:22
1  
@Marie.P.: ok, added to the answer. Is it clear now? –  Ilya Mar 22 '12 at 9:26
    
crystalclear! Awesome, thanks! –  Marie. P. Mar 22 '12 at 9:32

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.