Notation for derivatives: $u_{xx}$ versus $u''$

Question

If I have a problem in which I'm supposed to find $u(x)$ such that $u_{xx}(x) - 3u_x(x) + 2u(x) = 0$, is the equation the same as $u(x)''-3u(x)'+ 2u(x) = 0$, meaning that the first term, $u_{xx}(x)$, is the second derivative of $u(x)$ and the second term, $-3u_x$, is the product of $3$ and the first derivative of $u(x)$?

I'm not familiar with this specific notation and I'm a little stumped on where to find a description.

Answer 1

In my understanding, $u_x$ is when differentiating with respect to x, but $u'$ is whole derivative.

Answer 2

$u_x$ denotes a partial derivative with respect to $x$ - if $u$ is a function of only $x$, this is the same as the total derivative $\frac {du(x)}{dx}$.

If, however, $u$ is a function of $x$ and $y$, then the partial derivative is different.

For example, if $u(x,y) = 3x^2y$ , then $d/dx\; u(x,y)= 3 x \left(x \left(x y'\right)+2 y\right) , but \; u_x = 6xy. $

$u_{xx}$ is just the partial derivative applied twice.

the partial derivative with respect to x is a derivative where we treat other variables as constants.

Answer 3

Since the equation you talk about is ODE, usually $u_x$ denotes the derivative in $\mathbb{R}$.

Additional info: but some author denote $u_x$ the gradient of $u$ and $u_{xx}$ the Hessian of $u$. Therefore, you must read carefully on explanation on notations.