Thread: Tough Differential Equation Problem

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

and

The only thing i could think of is but that obviously doesnt satisfy the initial value. Any help or hints is greatly appreciated.

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

How about . Then .
And also

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....

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.

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:



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



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):






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:




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.