(dy)/(dx)=3y, and y=2 when x=1. Solve the equation. Choose 1 answer: (A) y=e^(3x-6) (B) y=2e^(3x+1) (C) y=2e^(3x+3) (D) y=2e^(3x-3)

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(dy)/(dx)=3y, and  y=2 when  x=1. Solve the equation. Choose 1 answer: (A)  y=e^(3x-6) (B)  y=2e^(3x+1) (C)  y=2e^(3x+3) (D)  y=2e^(3x-3)

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Recognize First-Order Linear Equation: Recognize that the given differential equation (dy)/(dx)=3y is a first-order linear differential equation that can be solved using separation of variables.Separate Variables: Separate the variables by dividing both sides by y and multiplying both sides by dx to get (1/y)dy = 3dx.Integrate Both Sides: Integrate both sides of the equation. The integral of (1/y)dy is ln|y|, and the integral of 3dx is 3x. So we have ln|y| = 3x + C, where C is the constant of integration.Solve for y: Solve for y by exponentiating both sides to get rid of the natural logarithm. This gives us |y| = e^(3x+C). Since y is positive (y=2 when x=1), we can drop the absolute value to get y = e^(3x+C).Use Initial Condition: Use the initial condition y=2 when x=1 to find the value of C. Substituting these values into y = e^(3x+C) gives us 2 = e^(3*1+C), which simplifies to 2 = e^(3+C).Solve for C: Solve for C by taking the natural logarithm of both sides. We get ln(2) = 3 + C, which simplifies to C = ln(2) - 3.Substitute C Value: Substitute the value of C back into the equation y = e^(3x+C) to get the particular solution. This gives us y = e^(3x + ln(2) - 3).Simplify Equation: Simplify the equation using properties of exponents. We know that e^(ln(2)) = 2, so we can rewrite the equation as y = 2e^(3x - 3).Compare with Choices: Compare the simplified equation with the answer choices to find the correct one. The equation y = 2e^(3x - 3) matches with choice (D) y=2e^(3x-3).

$\frac{d y}{d x}=3 y$ , and $y=2$ when $x=1$ . $\newline$ Solve the equation. $\newline$ Choose $1$ answer: $\newline$ (A) $y=e^{3 x-6}$ $\newline$ (B) $y=2 e^{3 x+1}$ $\newline$ (C) $y=2 e^{3 x+3}$ $\newline$ (D) $y=2 e^{3 x-3}$

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