Dear MHF members,

The problem is as follows.
Let $\displaystyle u(x):=\int_{0}^{\infty}\dfrac{\mathrm{e}^{-t}\sin(tx)}{\sqrt{t}}\mathrm{d}t$ and $\displaystyle v(x):=\int_{0}^{\infty}\dfrac{\mathrm{e}^{-t}\cos(tx)}{\sqrt{t}}\mathrm{d}t$.
Show that $\displaystyle u$ and $\displaystyle v$ are solutions of a system of first order.
Hence find explicit forms of $\displaystyle u$ and $\displaystyle v$.

In the following link you may see that $\displaystyle u$ and $\displaystyle v$ are functions in $\displaystyle C^{1}$.