# Thread:

1. ## Tough Differential Equation Problem

I need to find a function F such that it is continuous everywhere and

$y'(t)=F(y(t))$ and $y(0)=0$

The only thing i could think of is $y(t)=e^t$ but that obviously doesnt satisfy the initial value. Any help or hints is greatly appreciated.

2. ## Re: Tough Differential Equation Problem

So you're supposed to find both F and y(t) for which the given conditions are true, or ...?

How about $y(t)=\sin t$. Then $y(0)=\sin 0=0$.
And also $y'(t)=(\sin t)'=\cos t =\sqrt{1-\sin^2t}=\sqrt{1-(y(t))^2}=F(y(t)).$

3. ## Re: Tough Differential Equation Problem

I forgot to mention that I need to find a function F such that the initial value problem has infinitely many solutions. I don't know if that changes anything....

4. ## Re: Tough Differential Equation Problem

So to clear things up: I don't need to find a function y since this function F should take any function y and spit out its derivative i think.

5. ## Re: Tough Differential Equation Problem

The problem with this question is that the complexity of any derivative is arbitrary, for example, if we know that the function y is of the general form:

$\large%20y(t)=at+b$

... where a and b are constants. It is easy to create a function F that will reliably produce y's derivative, as in:

$\large%20F(t)=\frac{at}{at+b}$

However, even a slight change in the general form of y will upset everything and F will no longer function, while, as stated earlier the above function F produces the derivative of y, it will not produce the derivative of x (with constants a, b and c):

$\large%20y(t)=at+b$
$\large%20x(t)=(at+b)^c$

$\large%20G(t)=ac\times%20t^{1-\frac{1}{c}}$

However, a bit of manipulation will create a function that will work for both x and y, (above) function G, since they are quite similar. this won't work in all instances; consider the functions p and q, defined below:

$\large%20p(t)=\frac{(ln(t)+abt)^{at}}{t^a-t^{b}sin(ab)}$
$\large%20q(t)=\sqrt[t^{2}+ab]{ab\times%20cot(tsin(\frac{ab}{t}))}$

Creating a function that will produce the derivatives of both these functions is an enormous, if not impossible, task. So, my conclusion is, it is possible to create a function F that will produce the derivative of a function y, but the general form of y must first be known. Otherwise we'd have to consider an infinite number of combinations of roots, fractions, logarithms, trigonometric functions and lord knows what else - which from my humble high-school perspective - seems impossible.

Sorry. =( I hope someone else has a more positive answer.
{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "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": 0.9340835213661194, "perplexity_flag": "head"}