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.

TIFR 2019 GS Part A, Problem 19

Functions on differential equation

Moderate

Real Analysis, Bartle, Sherbert

Do you really need a hint? Try it first!

The map is clearly not injective as and maps to . Can you check about surjectivity?

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

This indeed happens, as we can solve the ODE by usual methods: since the solutions to are of the form , we try to look for general $f$ of the form . Then implies that , and of course is smooth if is.

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