`cos^(2)x (dy)/(dx)+y = tan x`. The solution of this diff. eqn. is ……….

Answer

To solve the differential equation \( \cos^2 x \frac{dy}{dx} + y = \tan x \), we can follow these steps:

### Step 1: Rewrite the equation
First, we rewrite the equation in a standard form. We divide the entire equation by \( \cos^2 x \):

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

Since \( \tan x = \frac{\sin x}{\cos x} \), we can express the right-hand side as:

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

### Step 2: Identify \( p(x) \) and \( q(x) \)
From the standard form \( \frac{dy}{dx} + p(x)y = q(x) \), we identify:

- \( p(x) = \frac{1}{\cos^2 x} = \sec^2 x \)
- \( q(x) = \sec^2 x \sin x \)

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

\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \sec^2 x \, dx} = e^{\tan x} \]

### Step 4: Multiply through by the integrating factor
We multiply the entire differential equation by the integrating factor:

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

### Step 5: Rewrite the left-hand side
The left-hand side can be rewritten as the derivative of a product:

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

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

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

The left-hand side simplifies to:

\[ e^{\tan x} y = \int e^{\tan x} \sec^2 x \sin x \, dx + C \]

### Step 7: Solve the integral on the right-hand side
To solve the integral on the right-hand side, we can use substitution. Let \( t = \tan x \), then \( dt = \sec^2 x \, dx \). The integral becomes:

\[ \int e^t \sin(\tan^{-1}(t)) \, dt \]

However, for simplicity, we can express the integral in terms of known functions or numerical methods.

### Step 8: Solve for \( y \)
After integrating, we can express \( y \):

\[ y = \frac{1}{e^{\tan x}} \left( \int e^{\tan x} \sec^2 x \sin x \, dx + C \right) \]

### Final Solution
The final solution to the differential equation is:

\[ y = e^{-\tan x} \left( \int e^{\tan x} \sec^2 x \sin x \, dx + C \right) \]