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Solve the PDE $2u_x(x,t) + 3u_t(x,t) = 0$ for all $(x,t) \in \mathbb{R}^2$ with $u(x,0) = f(x)$ for all $x \in \mathbb{R}$, where $f \in L^1 \cap \mathcal{C}^1$. Hint: Use Fourier transformation on the PDE and the initial condition.

I do not quite see what this hint is good for. Simply noting that the PDE is equivalent to $u_x + \frac{3}{2}u_t = 0$ reduces this problem basically to the advecation equation, which is easily solvable by the method of characteristics.

Am I missing something?

3nondatur
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    There is no Rule Given By The Gods that every equation shall only be solved in way way. – Mariano Suárez-Álvarez Nov 23 '22 at 02:14
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    You're not missing anything. Such an equation can also be solved using the Fourier transform (in particular using its shift property), which might be instructive the first time one is learning Fourier transform methods in PDEs. – messenger Nov 23 '22 at 02:16
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    Applying Fourier transform for $x$ and using its linear property and the fact that it maps $u_x$ to $k\xi\hat{u}$ ($k$ is a constant that varies on the definition of Fourier transform you're using) you get the equation $2k\hat{u}(\xi, t) + 3\hat{u}_t(\xi, t)=0$ which, for every $\xi$ is a ODE in $t$. Can you get it from here? – Ain't No O Nov 23 '22 at 02:32

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