It's very common to solve partial differential equations via "separable solution", in the following way. Say we have the wave equation,
$$u_t=u_{xx}.$$
We often solve this by assuming a form $u(x,t)=X(x)T(t)$, which gives
$$\frac{T_t}{T}=\frac{X_{xx}}{X}=\lambda,$$
where $\lambda$ is the separation constant, and the final solution looks something like
$$u(x,t)=C\exp (\lambda t -i\sqrt{\lambda} x)$$
However, it seems to me we could just as easily have tried the ansatz $u(x,t)=X(x)+T(t)$. Then our PDE would look like
$$T_t=X_{xx}=\lambda$$
And we would find
$$X(x)=\frac{1}{2}\lambda x^2 + C_2 x + C_3,\qquad T(t)=\lambda t + C_1.$$
Basically, a polynomial solution instead of an exponential one.
Is there any good reason why we often present the first way instead of the second? I guess there are "niceness" properties that the exponentials have, but polynomial solutions are nice in some ways too.
from Hot Weekly Questions - Mathematics Stack Exchange
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