Solution of differential equation `dy-sinxsiny dx=0` is

Options

A. `e^( cos x) ."" tan ""(y)/(2) =C `
B. ` e^( cos x ) . "" tan Y=C`
C. `cos x , tan Y=C `
D. ` cos x , sin y=C `

Answer

To solve the differential equation \( dy - \sin x \sin y \, dx = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ dy = \sin x \sin y \, dx \]

**Hint:** Move all terms involving \( dy \) to one side and terms involving \( dx \) to the other side.

### Step 2: Separating Variables
Next, we separate the variables:
\[ \frac{dy}{\sin y} = \sin x \, dx \]

**Hint:** To separate variables, divide both sides by \( \sin y \) and multiply both sides by \( dx \).

### Step 3: Integrating Both Sides
Now, we integrate both sides:
\[ \int \frac{dy}{\sin y} = \int \sin x \, dx \]

The left side integrates to:
\[ \int \frac{dy}{\sin y} = \log |\csc y - \cot y| + C_1 \]
And the right side integrates to:
\[ \int \sin x \, dx = -\cos x + C_2 \]

**Hint:** Remember to include the constant of integration when performing indefinite integrals.

### Step 4: Setting the Integrals Equal
Setting the results of the integrals equal gives us:
\[ \log |\csc y - \cot y| = -\cos x + C \]
where \( C = C_2 - C_1 \) is a new constant.

**Hint:** Combine the constants of integration into a single constant.

### Step 5: Exponentiating Both Sides
To eliminate the logarithm, we exponentiate both sides:
\[ |\csc y - \cot y| = e^{-\cos x + C} = e^C e^{-\cos x} \]
Let \( k = e^C \) (where \( k \) is a positive constant):
\[ |\csc y - \cot y| = k e^{-\cos x} \]

**Hint:** Use properties of exponents to simplify the expression.

### Step 6: Final Form
We can express the equation without the absolute value by considering the cases for \( y \):
\[ \csc y - \cot y = \pm k e^{-\cos x} \]

**Hint:** The absolute value can be dropped by considering both positive and negative cases.

### Conclusion
Thus, the solution to the differential equation \( dy - \sin x \sin y \, dx = 0 \) is:
\[ e^{\cos x} \tan\left(\frac{y}{2}\right) = C \]
where \( C \) is a constant.

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