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Find a solution to dA/dt=-6A if A(0)=6. A(t)=

Find A Solution To DAdt6A If A06 At class=

Sagot :

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]

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