# Elementary question in partial differentiation

Let's say we have a function of the form $f(x+vt)$ where $v$ is a constant and $x,t$ are independent variables. How is $\frac{\partial f}{\partial x} = \frac{1}{v}\frac{\partial f}{\partial t}$ equal to $f$?

If I let $u=x+vt$ then $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} = \frac{\partial f/\partial t}{\partial u/\partial t}\frac{\partial u}{\partial x}=\frac{1}{v}\frac{\partial f}{\partial t}$ but I cannot infer that $\frac{1}{v}\frac{\partial f}{\partial t} = f$ unless I assume the form of D'Alembert's Solution to be the harmonic (exponential). For the general solution I do not know how this was arrived at.

Edit: I still don't get it, as the context does not help. But I assume since it is a physics text, $f$ can be written as a Fourier series/integral of exponentials. Assuming that, the above holds.

-

You are correct. Where did you see it written that you can make this inference? – Chris Taylor Oct 10 '11 at 10:33
There is a function $f:\ s\mapsto f(s)$ of one variable and a function $u:\ (x,t)\mapsto f(x+vt)$ of two variables. All you can say is that $${\partial u\over\partial x}={1\over v}{\partial u\over\partial t}=f'(x+vt)\qquad \forall x, \ \forall t\ .$$ – Christian Blatter Oct 10 '11 at 12:11
Are you by any chance of south asian descent? Your username "kuch nahi" means "nothing" in some sense in urdu/hindi – Tyler Hilton Oct 14 '11 at 2:13
@Tyler Yes (hindi). As I wrote in chat an hour ago, it is a close approximation of my progress in mathematics. – kuch nahi Oct 14 '11 at 2:41

## 1 Answer

You are correct - you can't infer that $f'(x) = f(x)$ unless $f$ is exponential, i.e. if $f(x)=A\exp(x)$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 14, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9262173175811768, "perplexity_flag": "middle"}