3.8 Q-6

Question Statement

Solve the following differential equation:

$$\sin y \cdot \csc x \cdot \frac{dy}{dx} = 1$$

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Background and Explanation

This is a first-order differential equation. To solve it, we will use the method of separation of variables. We separate the terms involving $y$ on one side and the terms involving $x$ on the other side, then integrate both sides. Additionally, we need to recall the trigonometric identity $\csc x = \frac{1}{\sin x}$.

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Solution

1. Rewrite the equation using trigonometric identity:

   The equation involves $\csc x$, which is the reciprocal of $\sin x$. First, rewrite the equation:

$$\sin y \cdot \frac{1}{\sin x} \cdot \frac{dy}{dx} = 1$$

This simplifies to:

$$\frac{\sin y}{\sin x} \cdot \frac{dy}{dx} = 1$$

2. Separate the variables:

   Now, move the terms involving $y$ to one side and the terms involving $x$ to the other side:

$$\sin y , dy = \sin x , dx$$

3. Integrate both sides:

   Now, integrate both sides:

$$\int \sin y , dy = \int \sin x , dx$$

The integral of $\sin y$ with respect to $y$ is $-\cos y$, and the integral of $\sin x$ with respect to $x$ is $-\cos x$. So, we get:

$$-\cos y = -\cos x + C_1$$

4. Simplify the result:

   Rearranging the equation:

$$\cos y = \cos x + C$$

Where $C = -C_1$, is the constant of integration.

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Key Formulas or Methods Used

• Separation of Variables: Rearranged the equation to separate the terms involving $y$ and $x$.
• Trigonometric Identity: Used $\csc x = \frac{1}{\sin x}$ to simplify the equation.
• Integration: Integrated both sides with respect to their respective variables.

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Summary of Steps

1. Rewrite the equation using the identity $\csc x = \frac{1}{\sin x}$.
2. Separate the variables: $\sin y , dy = \sin x , dx$.
3. Integrate both sides: $-\cos y = -\cos x + C_1$.
4. Simplify the result: $\cos y = \cos x + C$.