3.8 Q-3

Question Statement

Solve the following differential equation:

$$y , dx + x , dy = 0$$

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Background and Explanation

This is a first-order homogeneous differential equation that can be solved using the method of separation of variables. The goal is to rearrange the terms so that all terms involving $x$ are on one side and all terms involving $y$ are on the other side. After this, we can integrate both sides to find the solution.

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Solution

1. Rearrange the equation:

   Starting with the given equation:

$$y , dx + x , dy = 0$$

Move $x , dy$ to the right-hand side:

$$y , dx = -x , dy$$

2. Separate the variables:

   To separate the variables, divide both sides of the equation by $x , y$:

$$\frac{1}{x} , dx = -\frac{1}{y} , dy$$

3. Integrate both sides:

   Now, integrate both sides:

$$\int \frac{1}{x} , dx = -\int \frac{1}{y} , dy$$

The left side integrates to $\ln(x)$, and the right side integrates to $-\ln(y)$, with a constant of integration $\ln(C)$:

$$\ln(x) = -\ln(y) + \ln(C)$$

4. Combine the logarithms:

   Combine the logarithmic terms on the left side:

$$\ln(x) + \ln(y) = \ln(C)$$

Using the property of logarithms that $\ln(a) + \ln(b) = \ln(ab)$, we get:

$$\ln(xy) = \ln(C)$$

5. Solve for $xy$:

   Exponentiate both sides to eliminate the logarithm:

$$xy = C$$

This is the general solution to the differential equation.

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Key Formulas or Methods Used

• Separation of variables: Rearranged the terms to separate variables $x$ and $y$.
• Integration: Integrated both sides of the equation.
• Properties of logarithms: Used the property $\ln(a) + \ln(b) = \ln(ab)$ to combine the logarithmic terms.

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Summary of Steps

1. Rearrange the equation: $y , dx = -x , dy$
2. Separate the variables: $\frac{1}{x} , dx = -\frac{1}{y} , dy$
3. Integrate both sides: $\ln(x) = -\ln(y) + \ln(C)$
4. Combine the logarithmic terms: $\ln(xy) = \ln(C)$
5. Exponentiate both sides: $xy = C$