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 :)
$\endgroup$1 Answer
$\begingroup$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:
homogeneous autonomous $$ u'(x)+u(x) = 0. $$
homogeneous non-autonomous $$ u''(x)+\color{red}{x}\cdot u(x) = 0 $$
non-homogeneous autonomous $$ u'(x)-2u(x) = \color{red}{1} $$
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.
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