Solved on Feb 25, 2024

Find the general solution for the differential equation dAdt=7A\frac{d A}{d t}=-7 A, where A(t)=Ce7tA(t)=Ce^{-7t}.

STEP 1

Assumptions
1. The function A(t)A(t) is differentiable with respect to time tt.
2. The differential equation given is dAdt=7A\frac{d A}{d t}=-7 A.
3. We are looking for the general form of the function A(t)A(t).

STEP 2

Recognize that the given differential equation is a first-order linear homogeneous differential equation, which can be solved using separation of variables.

STEP 3

Separate the variables AA and tt to opposite sides of the equation.
1AdA=7dt\frac{1}{A} dA = -7 dt

STEP 4

Integrate both sides of the equation with respect to their respective variables.
1AdA=7dt\int \frac{1}{A} dA = \int -7 dt

STEP 5

Perform the integration on both sides.
lnA=7t+C\ln |A| = -7t + C
Here, CC is the constant of integration.

STEP 6

Exponentiate both sides to solve for AA and eliminate the natural logarithm.
elnA=e7t+Ce^{\ln |A|} = e^{-7t + C}

STEP 7

Simplify the left side using the property elnx=xe^{\ln x} = x for x>0x > 0.
A=e7t+C|A| = e^{-7t + C}

STEP 8

Recognize that eCe^{C} is just another constant, which we can call C1C_1 (since CC is arbitrary). Also, since AA can be positive or negative, we can drop the absolute value and include the constant C1C_1 to account for the sign.
A(t)=C1e7tA(t) = C_1 e^{-7t}

STEP 9

Note that C1C_1 can be any real number, as it is the constant of integration. Therefore, the general form of the function that satisfies the differential equation is:
A(t)=Ce7tA(t) = C e^{-7t}
Where CC is a constant that can be determined if an initial condition is given.

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