3.8 Q-18

Question Statement

Solve the differential equation:

$$\left( e^x + e^{-x} \right) \frac{d y}{d x} = e^x - e^{-x}$$

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Background and Explanation

This is a first-order linear differential equation. We can solve it using the separation of variables method. The goal is to separate the terms involving $y$ on one side and the terms involving $x$ on the other, and then integrate both sides.

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Solution

1. Start with the given equation:

   The equation is:

$$\left( e^x + e^{-x} \right) \frac{d y}{d x} = e^x - e^{-x}$$

2. Separate the variables:

   To separate the variables, divide both sides of the equation by $e^x + e^{-x}$:

$$\frac{d y}{d x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$

3. Integrate both sides:

   Now, integrate both sides with respect to $x$:

$$y = \int \frac{e^x - e^{-x}}{e^x + e^{-x}} , dx$$

This can be simplified as:

$$y = \ln(e^x + e^{-x}) + C$$

4. Final solution:

   The general solution to the differential equation is:

$$y = \ln \left( e^x + e^{-x} \right) + C$$

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Key Formulas or Methods Used

• Separation of Variables: Rearranging the equation to isolate the terms involving $y$ on one side and the terms involving $x$ on the other.
• Integration of Rational Functions: Using logarithmic properties to integrate and simplify.

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Summary of Steps

1. Start with the equation: $\left( e^x + e^{-x} \right) \frac{d y}{d x} = e^x - e^{-x}$.
2. Separate the variables: $\frac{d y}{d x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$.
3. Integrate both sides: $y = \int \frac{e^x - e^{-x}}{e^x + e^{-x}} , dx$.
4. Simplify the result: $y = \ln \left( e^x + e^{-x} \right) + C$.