`dy/dx=-(cosx(3cosy-7sinx-3))/(siny(3sinx-7cosy+7))`

Options

A. `(cos y-sinx-1)^(2)(sinx+cosy-1)^(5)=C`
B. `(cosy-sinx-1)^(2)(sinx+cosy-1)^(3)=C`
C. `(cosy-sinx-1)^(2)(sinx+cosy-1)^(7)=C`
D. None of these

Answer

To solve the differential equation

\[ \frac{dy}{dx} = -\frac{\cos x (3 \cos y - 7 \sin x - 3)}{\sin y (3 \sin x - 7 \cos y + 7)}, \]

we will follow these steps:

### Step 1: Substitute Variables
Let \( t = \cos y \) and \( \mu = \sin x \). Then, we differentiate these substitutions:

\[ \frac{dt}{dy} = -\sin y \frac{dy}{dx} \quad \text{and} \quad \frac{d\mu}{dx} = \cos x. \]

### Step 2: Rewrite the Differential Equation
Substituting \( t \) and \( \mu \) into the differential equation gives:

\[ \sin y \, dy = -\cos x \cdot \frac{3t - 7\mu - 3}{3\mu - 7t + 7} \, dx. \]

### Step 3: Cross-Multiply
Cross-multiplying leads to:

\[ \sin y \, dy = -\cos x \cdot \frac{3t - 7\mu - 3}{3\mu - 7t + 7} \, dx. \]

### Step 4: Separate Variables
This can be rearranged to:

\[ \frac{dt}{d\mu} = -\frac{3t - 7\mu - 3}{3\mu - 7t + 7}. \]

### Step 5: Homogenize the Equation
To convert this into a homogeneous equation, we can set \( t = v\mu \) and differentiate:

\[ dt = v \, d\mu + \mu \, dv. \]

### Step 6: Substitute Back
Substituting back into the equation gives:

\[ \frac{dt}{d\mu} = \frac{v \, d\mu + \mu \, dv}{d\mu} = v + \mu \frac{dv}{d\mu}. \]

### Step 7: Rearrange and Simplify
This leads to:

\[ v + \mu \frac{dv}{d\mu} = -\frac{3v\mu - 7\mu^2 - 3}{3\mu - 7v + 7}. \]

### Step 8: Solve the Resulting Equation
We can now solve this equation by separating variables and integrating.

### Step 9: Integrate
Integrate both sides to find \( v \) in terms of \( \mu \):

\[ \int \frac{(3 - 7v)}{(v^2 - 1)} dv = \int \frac{7}{\mu} d\mu. \]

### Step 10: Combine Results
After integrating, we can express the solution in terms of \( y \) and \( x \) by substituting back \( t = \cos y \) and \( \mu = \sin x \).

### Final Step: Write the General Solution
The general solution will be in the form:

\[ \cos y - 1 - \sin^2 x \cdot (\cos y - 1 + \sin^5 x) = C, \]

where \( C \) is a constant.

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