# Is this licit or valid?

let,s suppose we have the differential equation:

$$g(x)=y+Dy+D^{2}y+D^{3}y+..................$$

D=d/dx, then this is equal to:

$$g(x)= \frac{1}{1-D}y$$ or inverting $$y=(1-D)g(x)$$

although this should be the solution it seems too easy to be true..but is that correct?..thanks.

HallsofIvy
Homework Helper
Well, let me see. If you start with a differential equation that is meaningless (or which, at least, you didn't bother to define) then, I guess, any thing you want to say is just as meaningless as the equation!

AKG
Homework Helper
He's defining a function g by

$$g(x) = y(x) + y'(x) + \sum_{k=2}y^{(k)}(x)$$

Assuming that series converges for each x, g is well-defined. Then, is it true that y(x) = g(x) - g'(x) (assuming g, as defined, is differentiable). This essentially asks if it is true that $g'(x) = y'(x) + \sum_{k=2}y^{(k)}(x)$, i.e. is the rule that 'the derivative of a sum the sum of the derivatives' extendible to the case where the sum is one of infinitely many functions?

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AKG
Homework Helper
$$g'(x) = \lim _{h\to 0}\frac{\sum_{k=0}y^{(k)}(x+h) - \sum_{k=0}y^{(k)}(x)}{h}$$

$$= \lim _{h\to 0}\frac{\lim _{n\to\infty}\sum_{k=0}^ny^{(k)}(x+h) - \lim _{n\to\infty}\sum_{k=0}^ny^{(k)}(x)}{h}$$

$$= \lim _{h\to 0}\frac{\lim _{n\to\infty}\sum_{k=0}^n\left (y^{(k)}(x+h) - y^{(k)}(x)\right )}{h}$$

$$= \lim _{h\to 0}\lim _{n\to\infty}\sum_{k=0}^n\frac{y^{(k)}(x+h) - y^{(k)}(x)}{h}$$

$$= \lim _{n\to\infty}\lim _{h\to 0}\sum_{k=0}^n\frac{y^{(k)}(x+h) - y^{(k)}(x)}{h}$$

$$= \lim _{n\to\infty}\sum_{k=0}^n\lim _{h\to 0}\left (\frac{y^{(k)}(x+h) - y^{(k)}(x)}{h}\right )$$

$$= \lim _{n\to\infty}\sum_{k=0}^n\lim _{h\to 0}y^{(k+1)}(x)$$

$$= \sum _{k=1}y^{(k)}(x)$$

Are any of these steps unjustified? The only lines I'm not sure about are line 3 (does it need absolute convergence) and line 5.

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HallsofIvy said:
Well, let me see. If you start with a differential equation that is meaningless (or which, at least, you didn't bother to define) then, I guess, any thing you want to say is just as meaningless as the equation!

Why should it be meaningless?..is a differential equation of infinite order..Euler Himself worked with this type of equations..for example to solve f(x+1)-f(x)=1 he make a Taylor expansion getting:

$$1= \sum_{n=0}^{\infty} \frac{y^{n}}{n!}$$ (1)

for wich he gets the solution (in a form of infinite series)

$$y= \sum_{n=0}^{\infty}a_{n} e^{i2\pi i x}$$

where he concludes that the a(n) must be chosen so the equation (1) is satisfied

matt grime