For x ∈ R, let the function y(x) be the solution of the differential equation dy/dx + 12y = cos(πx/1

Q: Wait, why isn't the function periodic? It has a cosine term in it!

That is a brilliant observation! While the cosine term is indeed periodic, the function $y(x)$ also contains a transient term, $C e^{-12x}$. As $x$ increases, this exponential term decays rapidly toward zero. Because the function is a sum of a periodic component and a decaying component, it never quite repeats its values perfectly. It is 'almost' periodic, but not strictly so.

Q: Why does the line $y = \beta$ intersect the curve infinitely many times?

Think about the long-term behavior. As $x \to \infty$, the transient exponential term $C e^{-12x}$ vanishes. The function $y(x)$ settles into a pure sinusoidal oscillation. A sine wave oscillates between its maximum and minimum values infinitely many times. Therefore, any horizontal line $y = \beta$ that lies within this amplitude range will be crossed by the wave infinitely often.

Q: Can we use the method of undetermined coefficients instead of the integrating factor?

Absolutely! You could assume a particular solution of the form $y_p = A \cos(\frac{\pi}{12}x) + B \sin(\frac{\pi}{12}x)$ and solve for $A$ and $B$. However, the integrating factor method is often more robust for first-order equations and provides a direct path to the general solution, including the constant $C$. Both methods are valid, but the integrating factor is a powerful tool to keep in your arsenal.

Question

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 12y = \cos \left( \frac{\pi}{12}x \right)$, $y(0) = 0$. Then, which of the following statements is/are TRUE?

Options:

(A) $y(x)$ is an increasing function

(B) $y(x)$ is a decreasing function

(C) There exists a real number $\beta$ such that the line $y = \beta$ intersects the curve $y = y(x)$ at infinitely many points

(D) $y(x)$ is a periodic function

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a differential equation; we are uncovering the hidden rhythm of a physical system.

When you look at the equation

$$
\frac{dy}{dx} + 12y = \cos \left( \frac{\pi}{12}x ight)
$$

do not see just symbols and variables. See a system that is being pushed by an external force—the cosine wave—while simultaneously trying to settle down, represented by the $12y$ term. This is the essence of a first-order linear differential equation, and it is a beautiful piece of mathematical architecture.

### Phase 1: The Integrating Factor—Our Magic Key

Every great journey begins with identifying the terrain. Our equation is in the standard form

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

Here, $P(x) = 12$ and $Q(x) = \cos \left( \frac{\pi}{12}x ight)$.

The integrating factor, or $IF$, is the key that unlocks this door. It is defined as $e^{\int P(x) dx}$. Substituting our constant $P(x) = 12$, we get

$$
IF = e^{\int 12 dx} = e^{12x}
$$

This $e^{12x}$ is the 'multiplier' that will transform our complex equation into a simple derivative of a product. When we multiply the entire equation by $e^{12x}$, the left side becomes the derivative of $y \cdot e^{12x}$. It is elegant, isn't it? The complexity collapses into a single, manageable derivative.

### Phase 2: The Dance of Integration

Now, we face the heart of the problem: $\int e^{12x} \cos \left( \frac{\pi}{12}x ight) dx$. This is where many students feel a shiver of fear. But fear not!

We have a standard formula for this exact integral:

$$
\int e^{ax} \cos(bx) dx = \frac{e^{ax}}{a^2 + b^2} [a \cos(bx) + b \sin(bx)]
$$

By identifying $a = 12$ and $b = \frac{\pi}{12}$, we can bypass the tedious process of integration by parts. Substituting these values, we obtain the integral. It is a long expression, yes, but it is just a sequence of algebraic steps. We are simply evaluating the interaction between the exponential growth and the trigonometric oscillation.

### Phase 3: Finding the Particular Solution

After integrating, we have our general solution:

$$
y(x) = K \left[ 12 \cos \left( \frac{\pi}{12}x ight) + \frac{\pi}{12} \sin \left( \frac{\pi}{12}x ight) ight] + C e^{-12x}
$$

But we are not finished. We have an initial condition: $y(0) = 0$. This is our anchor.

By plugging in $x = 0$, we find that $C = -12K$. This constant $C$ is the 'memory' of the system—it represents the initial state that influences the future trajectory. With $C$ determined, we have the complete, particular solution. Look at it closely. It is a combination of a steady-state oscillation and a transient decay.

### Phase 4: The Qualitative Revelation

Finally, let us analyze the behavior. Why is it not periodic? Because of the term $C e^{-12x}$.

As $x$ grows, this term dies away, but it is never zero for finite $x$. It prevents the function from ever repeating its values exactly. Why is it not monotonic? Because the sine and cosine terms force the function to oscillate.

And the most thrilling part? As $x 	o \infty$, the transient term vanishes, and the function becomes a pure sine wave. This means that for any horizontal line $y = \beta$ within the amplitude of this wave, the curve will cross it infinitely many times. You have just analyzed the long-term stability of a dynamic system. You have looked into the heart of the equation and understood its soul. Keep this intuition, and no differential equation will ever intimidate you again.

Animated Solution for Mathematics - Differential Equations: For $x \in R$ , let the function $y (x)$ be the solution of the differential equation $\frac{d y}{d x} + 12 y = cos (\frac{π}{12} x)$ , $y (0) = 0$ . Then, which of the following statements is/are TRUE?

Select Answer:

Visualized Solution

Identify Differential Equation

Calculate Integrating Factor

Multiply and Integrate

Standard Integral Formula

Evaluate the Integral

Write General Solution

Apply Initial Condition

Find Constant C

Analyze Function Behavior

Intersection at Infinity

The Sigma Insight: Linear Differential Equations

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Phase 4

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