Solution of diff. eqn. `(dy)/(dx) -(tany)/(1+x) = (1+x) e^(x)sec y` is …………

Answer

To solve the differential equation

\[ \frac{dy}{dx} - \frac{\tan y}{1+x} = (1+x)e^x \sec y, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we will rearrange the equation to isolate the secant term on the left-hand side:

\[ \frac{dy}{dx} - \frac{\tan y}{1+x} = (1+x)e^x \sec y. \]

We can rewrite \(\sec y\) as \(\frac{1}{\cos y}\), which gives us:

\[ \frac{dy}{dx} - \frac{\tan y}{1+x} = (1+x)e^x \frac{1}{\cos y}. \]

### Step 2: Multiply by \(\cos y\)
Next, we multiply through by \(\cos y\) to eliminate the secant:

\[ \cos y \frac{dy}{dx} - \frac{\sin y}{1+x} = (1+x)e^x. \]

### Step 3: Substitution
Now, we will make the substitution \(v = \sin y\). Therefore, we have:

\[ \cos y \frac{dy}{dx} = \frac{dv}{dx}. \]

Substituting this into our equation gives:

\[ \frac{dv}{dx} - \frac{v}{1+x} = (1+x)e^x. \]

### Step 4: Identifying the Linear Differential Equation
This is now a first-order linear differential equation in \(v\):

\[ \frac{dv}{dx} + p(x)v = q(x), \]

where \(p(x) = -\frac{1}{1+x}\) and \(q(x) = (1+x)e^x\).

### Step 5: Finding the Integrating Factor
The integrating factor \(I(x)\) is given by:

\[ I(x) = e^{\int p(x) \, dx} = e^{\int -\frac{1}{1+x} \, dx} = e^{-\ln(1+x)} = \frac{1}{1+x}. \]

### Step 6: Multiplying by the Integrating Factor
Now we multiply the entire differential equation by the integrating factor:

\[ \frac{1}{1+x} \frac{dv}{dx} - \frac{v}{(1+x)^2} = e^x. \]

### Step 7: Integrating Both Sides
The left-hand side can be rewritten as the derivative of a product:

\[ \frac{d}{dx}\left(\frac{v}{1+x}\right) = e^x. \]

Integrating both sides gives:

\[ \frac{v}{1+x} = e^x + C, \]

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

### Step 8: Solving for \(v\)
Now, we can solve for \(v\):

\[ v = (1+x)(e^x + C). \]

### Step 9: Back Substituting for \(y\)
Since \(v = \sin y\), we have:

\[ \sin y = (1+x)(e^x + C). \]

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

\[ \sin y = (1+x)(e^x + C). \]

---