3.8 Q-17

Question Statement

Solve the differential equation:

$$\operatorname{Sec} x + \tan y \frac{d y}{d x} = 0$$

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Background and Explanation

This is a first-order differential equation that can be solved using separation of variables. The key concept is to rewrite the equation such that all terms involving $y$ are on one side and all terms involving $x$ are on the other. After separating, the equation can be integrated on both sides.

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Solution

1. Start with the given equation:

   The equation is:

$$\operatorname{Sec} x + \tan y \frac{d y}{d x} = 0$$

2. Rearrange the equation:

   Move the term involving $\frac{d y}{d x}$ to one side:

$$\tan y \frac{d y}{d x} = -\operatorname{Sec} x$$

3. Separate the variables:

   Multiply both sides by $dx$ and rearrange to separate the $y$-terms and $x$-terms:

$$\tan y , dy = -\operatorname{Sec} x , dx$$

4. Integrate both sides:

   Now, integrate both sides:

   • The left side is the integral of $\tan y$:

$$\int \tan y , dy = -\ln |\cos y|$$

• The right side is the integral of $\operatorname{Sec} x$:

$$\int \operatorname{Sec} x , dx = \ln |\sec x + \tan x|$$

This gives us:

$$-\ln |\cos y| = \ln |\sec x + \tan x| + \ln |C|$$

5. Simplify the equation:

   Combine the logarithmic terms:

$$\ln |\cos y| = \ln |C (\sec x + \tan x)|$$

6. Exponentiate both sides to eliminate the logarithms:

$$|\cos y| = C (\sec x + \tan x)$$

7. Final solution:

   The equation simplifies to:

$$\cos y = C (\sec x + \tan x)$$

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Key Formulas or Methods Used

• Separation of Variables: Rearranging the equation to isolate terms involving $y$ on one side and terms involving $x$ on the other side.
• Integration of Standard Trigonometric Functions: Using standard integrals for $\tan y$ and $\sec x$.
• Logarithmic Properties: Using logarithmic rules to combine terms.

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Summary of Steps

1. Start with the equation: $\operatorname{Sec} x + \tan y \frac{d y}{d x} = 0$.
2. Rearrange to isolate the derivative: $\tan y \frac{d y}{d x} = -\operatorname{Sec} x$.
3. Separate the variables: $\tan y , dy = -\operatorname{Sec} x , dx$.
4. Integrate both sides: $\int \tan y , dy = \int -\operatorname{Sec} x , dx$.
5. Simplify the logarithmic terms: $\ln |\cos y| = \ln |C (\sec x + \tan x)|$.
6. Exponentiate both sides: $\cos y = C (\sec x + \tan x)$.