solve the differential equation (dy)/(dx)=-(e^(x))/(8)

Recognize Problem: Recognize the problem as an antiderivative problem. We are asked to find the function y(x) whose derivative with respect to x is given by -(e^(x))/(8).Write Integral: Write down the integral to solve for y. To find y, we need to integrate the given function with respect to x. So, we write the integral as: y = ∫ (-(e^(x))/(8)) dxPerform Integration: Perform the integration. The integral of e^x with respect to x is e^x, and since we have a constant coefficient of -1/8, we can pull it out of the integral. Thus, we get: y = -1/8 ∫ e^x dx y = -1/8 * e^x + C where C is the constant of integration.Write Final Answer: Write the final answer. The antiderivative of the function (dy)/(dx) = -(e^(x))/(8) is y = -1/8 * e^x + C.

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solve the differential equation (dy)/(dx)=-(e^(x))/(8)

Solve the differential equation $\frac{d y}{d x}=-\frac{e^{x}}{8}$

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Q. Solve the differential equation

\frac{d y}{d x}=-\frac{e^{x}}{8}

Recognize Problem: Recognize the problem as an antiderivative problem. $\newline$ We are asked to find the function $y(x)$ whose derivative with respect to $x$ is given by $-\frac{e^{x}}{8}$ .
Write Integral: Write down the integral to solve for $y$ . To find $y$ , we need to integrate the given function with respect to $x$ . So, we write the integral as: $y = \int \left(-\frac{e^{x}}{8}\right) dx$
Perform Integration: Perform the integration. $\newline$ The integral of $e^x$ with respect to $x$ is $e^x$ , and since we have a constant coefficient of $-\frac{1}{8}$ , we can pull it out of the integral. Thus, we get: $\newline$ $y = -\frac{1}{8} \int e^x dx$ $\newline$ $y = -\frac{1}{8} \cdot e^x + C$ $\newline$ where $C$ is the constant of integration.
Write Final Answer: Write the final answer. $\newline$ The antiderivative of the function $\frac{dy}{dx} = -\frac{e^{x}}{8}$ is $y = -\frac{1}{8} \cdot e^{x} + C$ .