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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