Answer
For each $x_{\iota}$,
$\displaystyle \frac{\partial}{\partial x_{i}}=a_{\iota}e^{a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}}$
and
$\displaystyle \frac{\partial^{2}}{\partial x_{i}^{2}}=a_{i}^{2}e^{a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}}$.
Then,
$\displaystyle \frac{\partial^{2}u}{\partial x_{1}^{2}}+\frac{\partial^{2}u}{\partial x_{2}^{2}}+\cdots+\frac{\partial^{2}u}{\partial x_{n}^{2}}=(a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2})e^{a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}}$
... use what has been given: $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=1$...
$=e^{a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}}$
$=u$
Work Step by Step
All steps are included in the answer.