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

Question

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

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Recognize First-Order Linear Homogeneous: Recognize that the given differential equation is a first-order linear homogeneous differential equation which can be solved using separation of variables.Separate Variables and Integrate: Separate the variables by dividing both sides by y and multiplying both sides by dx to get (1/y)dy = 3dx.Solve for Constant C: Integrate both sides of the equation. The integral of (1/y)dy is ln|y|, and the integral of 3dx is 3x. ∫(1/y)dy = ∫3dx ln|y| = 3x + C, where C is the constant of integration.Substitute and Find Particular Solution: Solve for the constant C using the initial condition y=2 when x=1. ln|2| = 3(1) + C ln(2) = 3 + C C = ln(2) - 3Exponentiate to Solve for y: Substitute the value of C back into the equation ln|y| = 3x + C to get the particular solution. ln|y| = 3x + ln(2) - 3Simplify the Equation: Exponentiate both sides to solve for y. e^(ln|y|) = e^(3x + ln(2) - 3) y = e^(3x) * e^(ln(2) - 3)Simplify the equation using properties of exponents. y = e^(3x) * (e^(ln(2)) / e^3) y = e^(3x) * (2 / e^3) 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) $y=2 e^{3 x-3}$ $\newline$ (C) $y=2 e^{3 x+3}$ $\newline$ (D) $y=2 e^{3 x+1}$

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