Solution of the diff. eqn. `dx+x dy = e^(-y) sec^(2)ydy` is ………..

Answer

To solve the differential equation \( dx + x \, dy = e^{-y} \sec^2 y \, dy \), we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We can divide the entire equation by \( dy \):

\[ \frac{dx}{dy} + x = e^{-y} \sec^2 y \]

### Step 2: Identify \( p(y) \) and \( q(y) \)
This is now in the standard form of a linear first-order differential equation:

\[ \frac{dx}{dy} + p(y) x = q(y) \]

where:
- \( p(y) = 1 \)
- \( q(y) = e^{-y} \sec^2 y \)

### Step 3: Find the integrating factor
The integrating factor \( \mu(y) \) is given by:

\[ \mu(y) = e^{\int p(y) \, dy} = e^{\int 1 \, dy} = e^y \]

### Step 4: Multiply through by the integrating factor
Now we multiply the entire differential equation by the integrating factor \( e^y \):

\[ e^y \frac{dx}{dy} + e^y x = e^y e^{-y} \sec^2 y \]

This simplifies to:

\[ e^y \frac{dx}{dy} + e^y x = \sec^2 y \]

### Step 5: Recognize the left-hand side as a derivative
The left-hand side can be recognized as the derivative of the product \( e^y x \):

\[ \frac{d}{dy}(e^y x) = \sec^2 y \]

### Step 6: Integrate both sides
Now we integrate both sides with respect to \( y \):

\[ \int \frac{d}{dy}(e^y x) \, dy = \int \sec^2 y \, dy \]

This gives us:

\[ e^y x = \tan y + C \]

where \( C \) is the constant of integration.

### Step 7: Solve for \( x \)
Finally, we solve for \( x \):

\[ x = e^{-y} (\tan y + C) \]

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

\[ x = e^{-y} (\tan y + C) \]

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