Find all infinitely differentiable functions \(f:\mathbb{R}\to\mathbb{R}\) that satisfy \[f(x)+f'(x)+f''(x)+\cdots =(f(x))^2\]

*Inspired by trumpeter*

Find all infinitely differentiable functions \(f:\mathbb{R}\to\mathbb{R}\) that satisfy \[f(x)+f'(x)+f''(x)+\cdots =(f(x))^2\]

*Inspired by trumpeter*

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TopNewest\[f(x) + f'(x) + f''(x) + \ldots = (f(x))^2 \]

Differentiating it and adding \(f(x)\) on both sides we get,

\[f(x) + f'(x) + f''(x) + \ldots = 2f(x) \times f'(x) +f(x) \]

\[\implies (f(x))^2 = 2f(x) \times f'(x) + f(x) \]

Case 1 :\(f'(x) = 0\) \(\implies f(x) = c\), where \(c\) is a constant. Substituting in the above equation gives \(c=0\) or \(c=1\). There are two solutions here and they are \(f(x) = 0\) and \(f(x) = 1\).Case 2 :\(f'(x) \neq 0\). For our convenience let us denote \(f(x)\) with \(y\).The above equation transfers into \(y^2 = 2y \dfrac{dy}{dx} +y\). But as we assumed that \(y \neq 0\).

It implies that \(y = 2 \dfrac{dy}{dx} +1\), which infers that \(\dfrac{dx}{2} = \dfrac{dy}{y-1}\). Integrating this both sides we get, \[\dfrac{x}{2} = ln (y-1) -ln(c)\] \[\implies y-1 = c \times e^{x/2}\] \[\implies y= c \times e^{x/2} +1\]

Now let us find out the value of \(c\).

Substitute this in the original equation, then we get

\[ (1+c e^{x/2}) + \dfrac{c}{2}e^{x/2} + \dfrac{c}{4}e^{x/2} + \ldots = (1+c e^{x/2})^2 = 1+2ce^{x/2} + c^2 e^{x} \]

\[c e^{x/2} (1+ \dfrac{1}{2} + \dfrac{1}{4} + \ldots ) = 2ce^{x/2} + c^2 e^x \\ 2c e^{x/2} = 2ce^{x/2} + c^2 e^x \\ c^2 e^x = 0\]

But \(e^{x} > 0\) for all \(x \in \mathbb{R}\). It implies that \(c^2 =0\), i.e. \(c=0\). But this implies that \(f(x) = 1\), a contradiction, as we assumed that \(f'(x) \neq 0\).

So, there are only two solutions i.e. \(f(x) =0\) and \(f(x) =1\).

Thank you @Pranshu Gaba , For making me realize my mistake. – Surya Prakash · 1 year, 5 months ago

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I think you have made an error in integration in Case 2. The power of \(e\) is \( + \frac{x}{2} \) and not \( - \frac{x}{2} \).

Also, I think you are missing one crucial step: When we substitute \( f( x) = c \cdot e^{\frac{x}{2} } + 1 \) in the given equation, we get

\[ 1 + 2 c \cdot e^{\frac{x}{2}} = 1 + 2c \cdot e^{\frac{x}{2} } + c^{2} \cdot e^{x} \]

This means \( c = 0\), and therefore \( f(x) = 1 \) for all values of \(x\).

Another solution is \( f(x) = 0 \) for all values of \(x\). These are the only solutions to the given equations.

In the end, we must not forget to substitute our solutions back in the equations to check for extraneous solutions. Also, we must check if the summation on the left hand side converges, otherwise our solution might not be valid. – Pranshu Gaba · 1 year, 5 months ago

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pointwise\( f(x) = 0 \) or \( (\ldots) = 0 \). It is not true that we only have the 2 cases of \( f(x) =0 \)everywhereOR \( ( \ldots ) = 0 \)everywhere.For example, it might be possible that the equation satisfies

\[ \begin{cases} f(x) = 0 & x \leq 0 \\ ( \ldots ) = 0 & 0 < x < 1 \\ f(x) = 0 & x \geq 1 \\ \end{cases} \]

Of course, there is no reason why our domain is split up into intervals.

Note: It is not clear to me why you set your cases as such. That seems extremely arbitrary / unmotivated. Similarly, you will need to consider possibilities of

\[ \begin{cases} f'(x) = 0 & x \leq 0 \\ f'(x) \neq 0 & 0 < x < 1 \\ f'(x) = 0 & x \geq 1 \\ \end{cases} \] – Calvin Lin Staff · 1 year, 5 months ago

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It is easy to see that

\[f\left(x\right)^2-2f\left(x\right)f'\left(x\right)=f\left(x\right)\]

Since \[2f\left(x\right)f'\left(x\right)=\frac{d}{dx}f\left(x\right)^2=f'\left(x\right)+f''\left(x\right)+f'''\left(x\right)...\]

Rearranging gives:

\[f'\left(x\right)=\frac{f\left(x\right)}{2}-\frac{1}{2}\]

Solving this differential equation yields \[f\left(x\right)=c\cdot e^{\frac{x}{2}}+1\], where \(c\) is a constant that comes with integration. – Julian Poon · 1 year, 5 months ago

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IE Your third line should have been \( f(x) \left[ f'(x) - \frac{f(x) } {2} + \frac{1}{2} \right] = 0 \), which makes \(f(x) = 0 \) for all \(x\) as a possible solution. – Calvin Lin Staff · 1 year, 5 months ago

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I think you are missing one crucial step: When we substitute \( f( x) = c \cdot e^{\frac{x}{2} } + 1 \) in the given equation, we get

\[ 1 + 2 c \cdot e^{\frac{x}{2}} = 1 + 2c \cdot e^{\frac{x}{2} } + c^{2} \cdot e^{x} \]

This means \( c = 0\), and therefore \( f(x) = 1 \) for all values of \(x\).

Another solution is \( f(x) = 0 \) for all values of \(x\). These are the only solutions to the given equations.

In the end, we must not forget to substitute our solutions back in the equations to check for extraneous solutions. Also, we must check if the summation on the left hand side converges, otherwise our solution might not be valid. – Pranshu Gaba · 1 year, 5 months ago

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\[\left(c^2e^x+1+2ce^{\frac{x}{2}}\right)-\left(c^2e^x+ce^{\frac{x}{2}}\right)=ce^{\frac{x}{2}}+1\]

And hence, I was sure of my answer. Except, I did not consider the f(x)=constant case – Julian Poon · 1 year, 5 months ago

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However, I was referring to the original equation \(f(x)+f'(x)+f''(x)+\cdots =(f(x))^2\). Try substituting \( f(x) = c\cdot e^{\frac{x}{2}}+1 \) in that equation, and you will see that only one value of \(c\) satisfies it, that is \(c = 0 \). – Pranshu Gaba · 1 year, 5 months ago

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– Julian Poon · 1 year, 5 months ago

Oh yes, yeah I just realised. ThanksLog in to reply

To everyone that begin their solutions by differentiating both sides: How do you justify that \(\frac{d}{dx} (f(x) + f'(x) + f''(x) + \ldots) = f'(x) + f''(x) + f'''(x) + \ldots\)? As far as I know, differentiating term-by-term is only justifiable for finite number of terms. – Ivan Koswara · 1 year, 5 months ago

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\[\lim_{n\to\infty}\sum_{i=0}^n f^{(i)}(x)=(f(x))^2\]

where \(f^{(i)}(x)\) denotes the \(i^{\textrm{th}}\) derivative of \(f(x)\) and \(f^{(0)}(x)=f(x)\).

Note that since \(n\) isn't dependent on the function variable \(x\), you can differentiate both sides to get,

\[\lim_{n\to\infty}\sum_{i=1}^{n+1} f^{(n)}(x)=2f(x)f^\prime(x)\]

Since \(n\to\infty\), we also have \(n+1\to\infty\) and you can rewrite it as,

\[2f(x)f^\prime(x)+f(x)=\lim_{n\to\infty}\sum_{i=0}^n f^{(i)}(x)=(f(x))^2\]

Of course, we assume that \(f^{(i)}(x)\) exists for all non-negative integers \(i\) to proceed and verify solutions in the end by checking it against the original equation. – Prasun Biswas · 1 year, 5 months ago

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What you have to justify is the interchange of limits:

\[ \lim \sum \frac{d}{dx} f_i = \lim \frac{d}{dx} \sum f_i = \frac{d}{dx} \lim \sum f_i \] – Calvin Lin Staff · 1 year, 5 months ago

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Is this reasoning correct,since derivatives' degrees would be less than that of \(f(x)\),and the degree of \([f(x)]^2\) would be two times that of \(f(x)\),the only solution could be when \((f(x)=k=\text{constant}\),hence all the derivatives would be \(0\) we would have,\(k=k^2 \Longrightarrow k=0,1\). This solution is for those functions of \(x\) which have only \(x\) and its powers(notified by @Surya Prakash ). – Adarsh Kumar · 1 year, 5 months ago

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In other words, you have (unnecessarily) restricted yourself to looking at finite degree polynomials, which is a very small subset of all functions. – Calvin Lin Staff · 1 year, 5 months ago

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– Adarsh Kumar · 1 year, 5 months ago

Ohk sir.Thanx for correcting me!Log in to reply

@Daniel Liu @Calvin Lin – Adarsh Kumar · 1 year, 5 months ago

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– Julian Poon · 1 year, 5 months ago

\(\left(\frac{d}{dx}f(x)\right)^{2} = \frac{d^{2}}{dx^{2}} f(x)^{2}\) is not true for most functions.Log in to reply

0,1 – Rui-Xian Siew · 1 year, 5 months ago

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– Calvin Lin Staff · 1 year, 5 months ago

Are those the only solutions? Why?Log in to reply

– Rui-Xian Siew · 1 year, 5 months ago

Haha I think the others had explained it in great detail. At first I just feel that f(X) should be constants, then I got the constants.Log in to reply