This equation is separable,
[tex]\dfrac{dA}{dt} = -6A \implies \dfrac{dA}A = -6\,dt[/tex]
Integrate both sides and solve for [tex]A[/tex] :
[tex]\displaystyle \int \frac{dA}A = -6 \int dt[/tex]
[tex]\ln|A| = -6t + C[/tex]
[tex]e^{\ln|A|} + e^{-6t+C}[/tex]
[tex]\implies A(t) = Ce^{-6t}[/tex]
Solve for [tex]C[/tex] using the initial value.
[tex]A(0) = 6 \implies 6 = Ce^0 \implies C=6[/tex]
Then the particular solution is
[tex]\boxed{A(t) = 6e^{-6t}}[/tex]
