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# Understand the problem

Consider maps $$C^{\infty} \to C^{\infty}$$ s.t $$f \mapsto f+ \frac{df}{dx}$$. We have to check whether this map is injective or surjective.

##### Source of the problem
TIFR 2019 GS Part A, Problem 19
##### Topic
Functions on differential equation
Moderate
##### Suggested Book
Real Analysis, Bartle, Sherbert
Do you really need a hint? Try it first!

The map is clearly not injective as $x$ and $x+e^{-x}$ maps to $x+1$. Can you check about surjectivity?

Checking surjectivity is the same as solving the ODE $f'+f = g$ for $f$ and seeing if you assume that $g$ is smooth, then $f$ is also smooth. Can you try now?

This indeed happens, as we can solve the ODE by usual methods: since the solutions to $f'+f=0$ are of the form $f(x) = Ce^{-x}$, we try to look for general $f$ of the form $f(x) = C(x)e^{-x}$. Then $g(x) = f'(x)+f(x) = C'(x)e^{-x}-C(x)e^{-x} + C(x)e^{-x} = C'(x)e^{-x}$ implies that $C(x) = \int e^xf(x)\,{\rm d}x$, and of course $f(x) = e^{-x}\int_0^x e^tg(t)\,{\rm d}t$ is smooth if $g$ is.

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