Dear MHF members,

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

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