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I have following two PDEs:

$\frac{\partial u_1'}{\partial t}+f(u_2')=\nu(\frac{\partial^2 u_1'}{\partial x_2^2}+\frac{\partial^2 u_1'}{\partial x_3^2})$

$u_2'\frac{\partial u_1'}{\partial x_2}+u_3'\frac{\partial u_1'}{\partial x_3}=0$

I already obtained $u_2'$ and $u_3'$ from other PDES wherer both $u_2'$ and $u_3'$ contain of in total 4 constants $C_1, C_2, C_3, C_4$, occuring from some integration. Now I have these two PDES to solve for $u_1'$. I used the first PDE to solve for $u_1'$ now I want to check if the second PDE is fullfilled, which it should be. However this is only the case if I choose some of the constant $C_i$ to be imaginary. I dont know if this makes much sense however, especially since $u_i$ represent velocity components.

Is it ok for constants of PDES to be imaginary even for a "real" physical problem?

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    $\begingroup$ I am concerned by your use of the words "constant of integration". Recall that when integrating PDEs, one obtains functions of integration. $\int x \,\mathrm{d}y = x y + C_1(x)$ has $C_1(x)$ as a function of integration. Why? Differentiate with respect to $y$ and $C_1(x)$ vanishes. $\endgroup$ Commented Nov 9, 2020 at 23:00
  • $\begingroup$ Without seeing the expressions of these $u$, it is hard to tell. $\endgroup$
    – user65203
    Commented Nov 9, 2020 at 23:05

1 Answer 1

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$$y''+y=0$$ has the general solution

$$y=c_+e^{ix}+c_-e^{-ix}.$$

With the initial conditions $y(0)=0,y'(0)=1$, $c_-+c_+=0$ and $i(c_+-c_-)=1$ and finally

$$y=-\frac i2e^{ix}+\frac i2e^{-ix}=\sin x.$$

This could answer your question.

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