(dy)/(dt)=2y, and y=8 when t=0. Solve the equation. Choose 1 answer: (A) y=e^(2t) (B) y=2e^(8t) (C) y=e^(8t) (D) y=8e^(2t)

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(dy)/(dt)=2y, and  y=8 when  t=0. Solve the equation. Choose 1 answer: (A)  y=e^(2t) (B)  y=2e^(8t) (C)  y=e^(8t) (D)  y=8e^(2t)

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Separation of Variables: The given differential equation is (dy)/(dt) = 2y. This is a first-order linear differential equation that can be solved using separation of variables. We will separate the variables y and t to integrate them separately.Integrating Separated Variables: We rewrite the equation to separate the variables: (1/y)dy = 2 dt. This allows us to integrate both sides with respect to their respective variables.Finding Constant of Integration: Integrating both sides, we get: ∫(1/y)dy = ∫2 dt. The integral of (1/y)dy is ln|y| and the integral of 2 dt is 2t. So we have ln|y| = 2t + C, where C is the constant of integration.Substitute Initial Condition: To find the constant of integration C, we use the initial condition given: y=8 when t=0. Substituting these values into the equation ln|y| = 2t + C, we get ln|8| = 2(0) + C. This simplifies to ln(8) = C.Final Solution: Now we have the constant C, which is ln(8). We substitute this back into the equation ln|y| = 2t + C to get ln|y| = 2t + ln(8).To solve for y, we exponentiate both sides of the equation to get rid of the natural logarithm: e^(ln|y|) = e^(2t + ln(8)). This simplifies to |y| = e^(2t) * e^(ln(8)).Since e^(ln(8)) is simply 8, we have |y| = 8e^(2t). Because y is positive (as given by the initial condition y=8 when t=0), we can drop the absolute value to get y = 8e^(2t).

$\frac{d y}{d t}=2 y$ , and $y=8$ when $t=0$ . $\newline$ Solve the equation. $\newline$ Choose $1$ answer: $\newline$ (A) $y=e^{2 t}$ $\newline$ (B) $y=2 e^{8 t}$ $\newline$ (C) $y=e^{8 t}$ $\newline$ (D) $y=8 e^{2 t}$

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dydt=2y \frac{d y}{d t}=2 y dtdy​=2y, and y=8 y=8 y=8 when t=0 t=0 t=0.\newlineSolve the equation.\newlineChoose 111 answer:\newline(A) y=e2t y=e^{2 t} y=e2t\newline(B) y=2e8t y=2 e^{8 t} y=2e8t\newline(C) y=e8t y=e^{8 t} y=e8t\newline(D) y=8e2t y=8 e^{2 t} y=8e2t

$\frac{d y}{d t}=2 y$ , and $y=8$ when $t=0$ . $\newline$ Solve the equation. $\newline$ Choose $1$ answer: $\newline$ (A) $y=e^{2 t}$ $\newline$ (B) $y=2 e^{8 t}$ $\newline$ (C) $y=e^{8 t}$ $\newline$ (D) $y=8 e^{2 t}$