Solution of diff. eqn. `"sec"x (dy)/(dx)=y+sin x` is ……….

Answer

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

### Step 1: Rewrite the equation
First, we rewrite the equation in a more manageable form. We can divide both sides by \( \sec x \):

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

### Step 2: Rearrange the equation
Next, we rearrange the equation to isolate \( \frac{dy}{dx} \):

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

This is now in the standard form of a first-order linear differential equation:

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

where \( P(x) = -\cos x \) and \( Q(x) = \sin x \cos x \).

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

\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int -\cos x \, dx} = e^{-\sin x} \]

### Step 4: Multiply through by the integrating factor
Now, we multiply the entire differential equation by the integrating factor:

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

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

\[ \frac{d}{dx}(e^{-\sin x} y) = e^{-\sin x} \sin x \cos x \]

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

\[ \int \frac{d}{dx}(e^{-\sin x} y) \, dx = \int e^{-\sin x} \sin x \cos x \, dx \]

The left-hand side simplifies to:

\[ e^{-\sin x} y = \int e^{-\sin x} \sin x \cos x \, dx + C \]

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

\[ \int e^{-t} t \, dt \]

This can be solved using integration by parts:

\[ \int e^{-t} t \, dt = -te^{-t} + \int e^{-t} \, dt = -te^{-t} - e^{-t} + C \]

Substituting back \( t = \sin x \):

\[ \int e^{-\sin x} \sin x \cos x \, dx = -\sin x e^{-\sin x} - e^{-\sin x} + C \]

### Step 8: Substitute back to find y
Now substituting back into our equation:

\[ e^{-\sin x} y = -\sin x e^{-\sin x} - e^{-\sin x} + C \]

Multiply through by \( e^{\sin x} \):

\[ y = -\sin x - 1 + Ce^{\sin x} \]

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

\[ y = -\sin x - 1 + Ce^{\sin x} \]

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