Let the solution curve y = y(x) of the differential equation (1 + e²ˣ)(dy/dx + y) = 1 pass through t

Q: Why can't I just separate the variables $x$ and $y$?

Great question! Separation of variables only works if you can write the equation as $f(x)dx = g(y)dy$. Here, the term $(\frac{dy}{dx} + y)$ is locked together. This is a classic Linear Differential Equation, which requires the Integrating Factor method to 'unlock' the derivative.

Q: How do I know when to use the Integrating Factor?

Whenever you see a first-order differential equation that looks like $\frac{dy}{dx} + P(x)y = Q(x)$, where $P$ and $Q$ are functions of $x$ only, the Integrating Factor is your best friend. It turns the left side into the derivative of a product: $\frac{d}{dx}(y \cdot I.F.)$.

Q: Why does $\tan^{-1}(e^x)$ approach $\frac{\pi}{2}$ as $x \to \infty$?

Think about the behavior of $e^x$. As $x$ goes to infinity, $e^x$ also goes to infinity. The function $\tan^{-1}(\theta)$ has a horizontal asymptote at $\frac{\pi}{2}$ as $\theta \to \infty$. So, as the input to the arctan function explodes, the output settles at $\frac{\pi}{2}$.

Question

Let the solution curve $y = y(x)$ of the differential equation $(1 + e^{2x})(\frac{dy}{dx} + y) = 1$ pass through the point $(0, \frac{\pi}{2})$. Then, $\lim_{x 	o \infty} e^x y(x)$ is equal to :

Options:

(A) $\frac{\pi}{4}$

(B) $\frac{3\pi}{4}$

(C) $\frac{\pi}{2}$

(D) $\frac{3\pi}{2}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler in the world of calculus. Today, we are going to dismantle a problem that, at first glance, might seem like a tangled mess of exponentials and derivatives.

In the realm of JEE Advanced, the most intimidating equations are often just simple concepts wearing a disguise. Our target is the differential equation $(1 + e^{2x})(\frac{dy}{dx} + y) = 1$.

### Phase 1: The Standard Form

When you see a differential equation, your first instinct should be to classify it. The structure $\frac{dy}{dx} + y$ is a massive hint that this is a Linear Differential Equation (LDE).

To make this explicit, we divide the entire equation by $(1 + e^{2x})$:

$$\frac{dy}{dx} + y = \frac{1}{1 + e^{2x}}$$

Now, compare this to the standard LDE form: $\frac{dy}{dx} + P(x)y = Q(x)$. Here, $P(x) = 1$ and $Q(x) = \frac{1}{1 + e^{2x}}$.

### Phase 2: The Magic of the Integrating Factor

We need the key to unlock this machine: the Integrating Factor (I.F.). The formula is elegant and powerful: $I.F. = e^{\int P(x) dx}$.

Since our $P(x)$ is just $1$, the integral is trivial: $\int 1 dx = x$. Thus, our Integrating Factor is $e^x$.

When we multiply the entire equation by $e^x$, the left-hand side collapses into the derivative of a product: $\frac{d}{dx}(y \cdot e^x)$. This turns a sum of terms into a single, manageable derivative.

### Phase 3: The Integration

Multiplying our equation by $e^x$, we obtain:

$$y \cdot e^x = \int \frac{e^x}{1 + e^{2x}} dx + C$$

Note that $e^{2x}$ is simply $(e^x)^2$. If we let $t = e^x$, then $dt = e^x dx$, and the integral transforms into $\int \frac{1}{1 + t^2} dt$.

This is a standard integral resulting in $	an^{-1}(t)$. Substituting back, we get:

$$y \cdot e^x = 	an^{-1}(e^x) + C$$

### Phase 4: Finding the Constant

We have the general solution, but we need the specific curve that passes through $(0, \frac{\pi}{2})$. Plugging in $x=0$ and $y=\frac{\pi}{2}$:

$$\frac{\pi}{2} \cdot e^0 = 	an^{-1}(e^0) + C$$

Since $e^0 = 1$ and $	an^{-1}(1) = \frac{\pi}{4}$, the equation becomes $\frac{\pi}{2} = \frac{\pi}{4} + C$. Solving for the constant, we find $C = \frac{\pi}{4}$.

Our particular solution is defined as:

$$y \cdot e^x = 	an^{-1}(e^x) + \frac{\pi}{4}$$

### Phase 5: The Final Crescendo

The question asks for $\lim_{x 	o \infty} e^x y(x)$. We have the expression for $e^x y(x)$ ready for evaluation.

We take the limit as $x 	o \infty$ of $	an^{-1}(e^x) + \frac{\pi}{4}$. As $x$ approaches infinity, $e^x$ approaches infinity, and $	an^{-1}(\infty)$ approaches $\frac{\pi}{2}$.

Therefore, the final limit is:

$$\frac{\pi}{2} + \frac{\pi}{4} = \mathbf{\frac{3\pi}{4}}$$

Animated Solution for Mathematics - Differential Equations: Let the solution curve $y = y (x)$ of the differential equation $(1 + e^{2 x}) (\frac{d y}{d x} + y) = 1$ pass through the point $(0, \frac{π}{2})$ . Then, $lim_{x \to \infty} e^{x} y (x)$ is equal to :

Select Answer:

Visualized Solution

Analyze the Differential Equation

Rearrange to Standard LDE Form

Identify $P (x)$ and $Q (x)$

Calculate the Integrating Factor (I.F.)

Set up the General Solution

Simplify the Integral Expression

Solve the Integral using Substitution

Apply Inverse Trig Formula

Apply Initial Condition

Solve for Constant $C$

Finalize the Value of $C$

Write the Final Equation

Set Up the Limit

Evaluate the Limit

Final Answer

The Sigma Insight: Linear Differential Equations

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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