Question Statement

Solve the following differential equation: $\frac{dy}{dx}=-\frac{1}{e^{3x}}$

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Background and Explanation

This is a first-order ordinary differential equation that can be solved using the method of separation of variables. To find the general solution, we rearrange the equation to isolate the variables $y$ and $x$ on opposite sides and then integrate both sides.

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Solution

To solve the differential equation, we follow these steps:

1. Separate the variables First, we rewrite the equation to separate the $dy$ and $dx$ terms. We also use the property of exponents $\frac{1}{e^a}=e^{-a}$ to simplify the expression: $\begin{array}{rl}\frac{dy}{dx}& =\frac{-1}{e^{3x}}\\ dy& =\frac{-1}{e^{3x}}dx\\ dy& =-e^{-3x}dx\end{array}$

2. Integrate both sides Now, we apply the integral sign to both sides of the equation: $\int dy=-\int e^{-3x}dx$

3. Evaluate the integrals The integral of $1$ with respect to $y$ is $y$. For the right side, we use the rule $\int e^{ax}dx=\frac{1}{a}{e}^{ax}+C$: $y=\frac{-e^{-3x}}{-3}+c_1$ Simplifying the signs, we get: $y=\frac{1}{3}{e}^{-3x}+c_1$

4. Final simplification To clear the fraction, we multiply the entire equation by $3$: $3y=e^{-3x}+3c_1$ Since $3c_1$ is just another constant, we can replace it with a single constant $c$: $3y=e^{-3x}+c(\text{where }c=3c_1)$

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Key Formulas or Methods Used

• Separation of Variables: A method to solve differential equations by moving all terms involving $y$ to one side and all terms involving $x$ to the other.
• Exponential Rule of Integration: $\int e^{ax}dx=\frac{1}{a}{e}^{ax}+C$.
• Exponent Property: $\frac{1}{x^n}=x^{-n}$.

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Summary of Steps

1. Rewrite the fraction $\frac{1}{e^{3x}}$ as $e^{-3x}$.
2. Move $dx$ to the right side to separate the variables.
3. Integrate both sides, remembering to divide by the coefficient of $x$ (which is $-3$).
4. Multiply by $3$ and simplify the constant of integration to reach the final general solution.