Find the general solution of the following differential equations :
`(dx)/(dy)+x=tan y + sec^(2)y`.

Answer

To solve the differential equation

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

we will follow these steps:

### Step 1: Identify the form of the equation
The given equation is in the standard form of a first-order linear differential equation:

\[ \frac{dx}{dy} + P(y)x = Q(y), \]

where \( P(y) = 1 \) and \( Q(y) = \tan y + \sec^2 y \).

### Step 2: 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 3: Multiply the entire equation by the integrating factor
Now, we multiply the entire differential equation by the integrating factor \( e^y \):

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

### Step 4: Rewrite the left-hand side
The left-hand side can be rewritten as the derivative of a product:

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

### Step 5: Integrate both sides
Now, we integrate both sides with respect to \( y \):

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

The left-hand side simplifies to:

\[ e^y x = \int e^y (\tan y + \sec^2 y) \, dy. \]

### Step 6: Solve the right-hand side integral
To solve the integral on the right-hand side, we can split it into two parts:

\[ \int e^y \tan y \, dy + \int e^y \sec^2 y \, dy. \]

The second integral can be solved directly:

\[ \int e^y \sec^2 y \, dy = e^y \tan y + C_1, \]

where \( C_1 \) is a constant of integration.

The first integral, \( \int e^y \tan y \, dy \), can be solved using integration by parts or a table of integrals, yielding a more complex expression. However, for simplicity, we can denote it as \( I(y) \).

### Step 7: Combine results
Thus, we have:

\[ e^y x = I(y) + e^y \tan y + C, \]

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

### Step 8: Solve for \( x \)
Finally, we solve for \( x \):

\[ x = \frac{I(y) + e^y \tan y + C}{e^y}. \]

### Final General Solution
The general solution of the differential equation is:

\[ x = \frac{I(y)}{e^y} + \tan y + Ce^{-y}. \]