Find the integrating factor of the differential equation (1 + tan y) (dx - dy) + 2x dy=0.

Answer

To find the integrating factor of the differential equation \((1 + \tan y)(dx - dy) + 2x dy = 0\), we can follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the given differential equation in a more standard form. The equation can be rearranged as:
\[ (1 + \tan y) dx + (2x - 1) dy = 0 \]

### Step 2: Identify \(P\) and \(Q\)
From the equation, we identify:
- \(P = 1 + \tan y\)
- \(Q = 2x - 1\)

### Step 3: Check for Exactness
To check if the differential equation is exact, we need to compute \(\frac{\partial Q}{\partial x}\) and \(\frac{\partial P}{\partial y}\):
\[ \frac{\partial Q}{\partial x} = 2 \]
\[ \frac{\partial P}{\partial y} = \sec^2 y \]
Since \(\frac{\partial Q}{\partial x} \neq \frac{\partial P}{\partial y}\), the equation is not exact.

### Step 4: Find the Integrating Factor
Since the equation is not exact, we need to find an integrating factor. For a differential equation of the form \(P dx + Q dy = 0\), if we can express it in the form:
\[ \frac{dx}{dy} + P(x,y) = Q(x,y) \]
we can find an integrating factor. Rearranging gives:
\[ \frac{dx}{dy} = -\frac{Q}{P} = -\frac{2x - 1}{1 + \tan y} \]

### Step 5: Identify \(p\) and \(q\)
From the rearranged equation, we have:
\[ p = -\frac{2x - 1}{1 + \tan y} \]
This can be simplified to:
\[ p = \frac{2x - 1}{1 + \tan y} \]

### Step 6: Find the Integrating Factor
For a linear differential equation of the form:
\[ \frac{dx}{dy} + p(x,y) = q(y) \]
the integrating factor is given by:
\[ IF = e^{\int p(y) dy} \]
In our case, we have:
\[ p(y) = \frac{2}{1 + \tan y} \]
Thus, the integrating factor becomes:
\[ IF = e^{\int \frac{2}{1 + \tan y} dy} \]

### Step 7: Solve the Integral
To solve the integral:
\[ \int \frac{2}{1 + \tan y} dy \]
We can use the substitution \(u = \tan y\), which gives \(du = \sec^2 y dy\). Therefore:
\[ dy = \frac{du}{\sec^2 y} = \frac{du}{1 + u^2} \]
Thus, the integral becomes:
\[ \int \frac{2}{1 + u} \cdot \frac{du}{1 + u^2} \]

### Step 8: Finalizing the Integrating Factor
After evaluating the integral, we can express the integrating factor in a usable form. The final integrating factor will be:
\[ IF = e^{\text{result of the integral}} \]