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

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\frac{dy}{dx} = -4y

, and

y=3

when

x=2

. Solve the equation.

Identify Type and Method: Identify the type of differential equation and the method to solve it. $\newline$ The given differential equation $\frac{dy}{dx} = -4y$ 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. $\newline$ The general solution to the differential equation $\frac{dy}{dx} = -4y$ can be found by separating variables. We can rewrite the equation as $\frac{dy}{y} = -4 dx$ and then integrate both sides.
Perform Integration: Perform the integration on both sides. $\newline$ $\int(\frac{1}{y}) dy = \int(-4) dx$ $\newline$ $\ln|y| = -4x + C$ , where $C$ is the constant of integration.
Solve for y: Solve for y. $\newline$ To solve for y, we exponentiate both sides of the equation: $\newline$ $e^{\ln|y|} = e^{(-4x + C)}$ $\newline$ $y = e^{(-4x)} \cdot e^{C}$ $\newline$ Since $e^{C}$ is just a constant, we can rename it as $C'$ : $\newline$ $y = C'e^{(-4x)}$
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 $C'$ : $3 = C'e^{(-4\cdot2)}$ $3 = C'e^{(-8)}$ $C' = \frac{3}{e^{(-8)}}$
Calculate $C'$ Value: Calculate the value of $C'$ . $C' = \frac{3}{e^{-8}} = 3 \times e^8$
Write Final Solution: Write the final solution with the value of $C'$ . $\newline$ The final solution to the differential equation with the initial condition is: $\newline$ $y = (3 \cdot e^8)e^{-4x}$