(dy)/(dt)=-7y, and y=3 when t=0. Solve the equation. Choose 1 answer: (A) y=3e^(-7t) (B) y=e^(-7t) (C) y=3+e^(-7t) (D) y=2+e^(-7t)

Question

(dy)/(dt)=-7y, and  y=3 when  t=0. Solve the equation. Choose 1 answer: (A)  y=3e^(-7t) (B)  y=e^(-7t) (C)  y=3+e^(-7t) (D)  y=2+e^(-7t)

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Identify Differential Equation: The given differential equation is a first-order linear homogeneous differential equation. To solve it, we can use separation of variables.Separate Variables: Separate the variables y and t by dividing both sides by y and multiplying both sides by dt: (dy/y) = -7 dtIntegrate Equations: Integrate both sides of the equation: ∫(1/y) dy = ∫-7 dtFind Constant of Integration: The integral of 1/y with respect to y is ln|y|, and the integral of -7 with respect to t is -7t. So we have: ln|y| = -7t + C, where C is the constant of integration.Apply Initial Condition: To find the constant of integration C, we use the initial condition y=3 when t=0: ln|3| = -7(0) + C C = ln(3)Finalize Equation: Now we have the equation with the constant C: ln|y| = -7t + ln(3)Exponentiate to Eliminate Log: To solve for y, we exponentiate both sides of the equation to get rid of the natural logarithm: e^(ln|y|) = e^(-7t + ln(3))Remove Absolute Value: Since e^(ln|x|) = |x| for any x, we have: |y| = e^(-7t) * e^(ln(3))Final Solution: Since y is initially positive (y=3 when t=0), we can drop the absolute value bars: y = 3 * e^(-7t)The solution to the differential equation is y = 3 * e^(-7t), which corresponds to answer choice (A).

$\frac{d y}{d t}=-7 y$ , and $y=3$ when $t=0$ . $\newline$ Solve the equation. $\newline$ Choose $1$ answer: $\newline$ (A) $y=3 e^{-7 t}$ $\newline$ (B) $y=e^{-7 t}$ $\newline$ (C) $y=3+e^{-7 t}$ $\newline$ (D) $y=2+e^{-7 t}$

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