Solution of the diff. eqn. `siny (dy//dx)= cos y (1-x cos y)` is ……….

Answer

To solve the differential equation given by

\[ \sin y \frac{dy}{dx} = \cos y (1 - x \cos y), \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate the terms involving \(y\) and \(x\):

\[ \sin y \frac{dy}{dx} = \cos y - x \cos^2 y. \]

### Step 2: Dividing by \(\cos^2 y\)
Next, we divide both sides by \(\cos^2 y\):

\[ \frac{\sin y}{\cos^2 y} \frac{dy}{dx} = \frac{\cos y}{\cos^2 y} - x. \]

This simplifies to:

\[ \tan y \frac{dy}{dx} = \sec y - x. \]

### Step 3: Substituting \(v = \sec y\)
We will now use the substitution \(v = \sec y\). Therefore, we have:

\[ \frac{dy}{dx} = \frac{dv}{dx} \cdot \frac{1}{\sec y \tan y} = \frac{dv}{dx} \cdot \frac{1}{v \sqrt{v^2 - 1}}. \]

### Step 4: Substitute into the Equation
Substituting \( \frac{dy}{dx} \) into our rearranged equation gives us:

\[ \tan y \cdot \frac{1}{v \sqrt{v^2 - 1}} \frac{dv}{dx} = v - x. \]

### Step 5: Rearranging the Equation
Rearranging gives us:

\[ \frac{dv}{dx} = (v - x) v \sqrt{v^2 - 1}. \]

### Step 6: Separating Variables
Now we separate the variables:

\[ \frac{dv}{(v - x) v \sqrt{v^2 - 1}} = dx. \]

### Step 7: Integrating Both Sides
Next, we integrate both sides. The left side requires partial fraction decomposition and trigonometric substitution, while the right side integrates directly to \(x + C\).

### Step 8: Solving the Integrals
After performing the integrals, we will have:

\[ \text{LHS} = \text{some function of } v, \]
and
\[ \text{RHS} = x + C. \]

### Step 9: Back Substituting for \(y\)
Finally, we will back substitute \(v = \sec y\) to express the solution in terms of \(y\):

\[ \sec y = \text{result from LHS} + x + C. \]

### Final Solution
Thus, the solution of the differential equation is:

\[ \sec y = x + 1 + Ce^x. \]