 Solution

Put $u=a+i\omega t$. ThenThe $G$ equation from part (b) is satisfied whensoWith this value, the $F$ equation follows from ${\tanh }^{2}u=1-{\mathrm{sech}}^{2}u$: both its constant and ${\mathrm{sech}}^{2}u$ coefficients agree. Thus the stated $F,G$ satisfy both shape equations; the remaining scalar equation determines the normalization factor $N(t)$.