# Thread: Tough Differential Equation Problem

1. ## Tough Differential Equation Problem

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

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

The only thing i could think of is $\displaystyle y(t)=e^t$ 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 $\displaystyle y(t)=\sin t$. Then $\displaystyle y(0)=\sin 0=0$.
And also $\displaystyle y'(t)=(\sin t)'=\cos t =\sqrt{1-\sin^2t}=\sqrt{1-(y(t))^2}=F(y(t)).$

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:

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

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

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

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):

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

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

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:

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

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.