dy/dx = -4y, and y=3 when x=2. Solve the equation.

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Identify Type and Method: Identify the type of differential equation and the method to solve it. The given differential equation dy/dx = -4y is a first-order linear homogeneous differential equation. The method to solve such an equation is to find the integrating factor and then integrate both sides.Solve Differential Equation: Solve the differential equation. The general solution to the differential equation dy/dx = -4y can be found by separating variables. We can rewrite the equation as dy/y = -4 dx and then integrate both sides.Perform Integration: Perform the integration on both sides. ∫(1/y) dy = ∫(-4) dx ln|y| = -4x + C, where C is the constant of integration.Solve for y: Solve for y. To solve for y, we exponentiate both sides of the equation: e^(ln|y|) = e^(-4x + C) y = e^(-4x) * e^C Since e^C is just a constant, we can rename it as C': y = C'e^(-4x)Use Initial Condition: Use the initial condition to find the value of C'. We are given that y=3 when x=2. We substitute these values into the equation to solve for C': 3 = C'e^(-4*2) 3 = C'e^(-8) C' = 3/e^(-8)Calculate C' Value: Calculate the value of C'. C' = 3/e^(-8) = 3 * e^8Write Final Solution: Write the final solution with the value of C'. The final solution to the differential equation with the initial condition is: y = (3 * e^8)e^(-4x)

$\frac{dy}{dx} = -4y$ , and $y=3$ when $x=2$ . Solve the equation.

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dydx=−4y\frac{dy}{dx} = -4ydxdy​=−4y, and y=3y=3y=3 when x=2x=2x=2. Solve the equation.

$\frac{dy}{dx} = -4y$ , and $y=3$ when $x=2$ . Solve the equation.