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

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

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Recognize Linear Homogeneous Differential Equation: Recognize that the given differential equation is a first-order linear homogeneous differential equation which can be solved using separation of variables.Separate and Integrate Variables: Separate the variables by dividing both sides by y and multiplying both sides by dt to get dy/y = 2 dt.Solve for Constant C: Integrate both sides of the equation. The integral of 1/y dy is ln|y|, and the integral of 2 dt is 2t. So, ∫(1/y) dy = ∫2 dt leads to ln|y| = 2t + C, where C is the constant of integration.Calculate Value of C: Solve for the constant C using the initial condition y=8 when t=0. Plugging in the values, we get ln|8| = 2*0 + C, which simplifies to ln(8) = C.Substitute C into Equation: Calculate the value of C using the natural logarithm of 8. C = ln(8) = ln(2^3) = 3ln(2).Exponentiate to Solve for y: Substitute the value of C back into the equation ln|y| = 2t + C to get ln|y| = 2t + 3ln(2).Simplify Exponential Expression: Exponentiate both sides to solve for y. We get |y| = e^(2t + 3ln(2)).Drop Absolute Value for Final Solution: Simplify the right side using properties of exponents. e^(3ln(2)) is the same as (e^(ln(2)))^3, which simplifies to 2^3 or 8. So, |y| = e^(2t) * 8.Since y is positive for t=0 and y=8, 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) $y=e^{8 t}$ $\newline$ (B) $y=e^{2 t}$ $\newline$ (C) $y=8 e^{2 t}$ $\newline$ (D) $y=2 e^{8 t}$

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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=e8t y=e^{8 t} y=e8t\newline(B) y=e2t y=e^{2 t} y=e2t\newline(C) y=8e2t y=8 e^{2 t} y=8e2t\newline(D) y=2e8t y=2 e^{8 t} y=2e8t

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