# Imaginary integration constants

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?

• 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. Commented Nov 9, 2020 at 23:00
• Without seeing the expressions of these $u$, it is hard to tell.
– user65203
Commented Nov 9, 2020 at 23:05

$$y''+y=0$$ 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
$$y=-\frac i2e^{ix}+\frac i2e^{-ix}=\sin x.$$