If y = y(x) is the solution of the differential equation dy/dx + 2y = sin(2x), y(0) = 3/4, then y(π/

Q: Why can't I just use separation of variables to solve this?

Separation of variables is a fantastic technique, but it only works when you can completely isolate the $y$ terms from the $x$ terms. In this equation, the $2y$ term is added to the derivative $\frac{dy}{dx}$, which makes it impossible to separate them. This is a classic linear differential equation, and the Integrating Factor method is the standard, robust way to handle this structure.

Q: I keep getting lost in the integration by parts. Is there a faster way?

You are not alone! Integration by parts is prone to sign errors, especially with exponential and trigonometric functions. I highly recommend memorizing the standard integral formula $\int e^{ax} \sin(bx) dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx)$. It is a massive time-saver in competitive exams like JEE and significantly reduces the chance of making a mistake.

Q: Why does the first term vanish at $x = \frac{\pi}{8}$?

That is the beauty of this problem! When you substitute $x = \frac{\pi}{8}$ into the term $(\sin 2x - \cos 2x)$, the argument becomes $2 \cdot \frac{\pi}{8} = \frac{\pi}{4}$. Since $\sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$, their difference is exactly zero. It is a deliberate, elegant simplification that rewards your precision.

Question

If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} + 2y = \sin(2x), y(0) = \frac{3}{4}$, then $y(\frac{\pi}{8})$ is equal to:

Options:

(A) $e^{\pi/8}$

(B) $e^{\pi/4}$

(C) $e^{-\pi/4}$

(D) $e^{-\pi/8}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the graph of a function $y(x)$. You are not just looking at a static line; you are looking at a dynamic relationship defined by the differential equation:

$$\frac{dy}{dx} + 2y = \sin(2x)$$

This equation tells us how the slope of the curve, $\frac{dy}{dx}$, is intrinsically linked to the function's value $y$ and the oscillating nature of $\sin(2x)$. Our mission is to find the specific curve that satisfies this relationship and passes through the point $(0, \frac{3}{4})$.

### The Magic Multiplier: The Integrating Factor

When we look at $\frac{dy}{dx} + 2y = \sin(2x)$, we immediately recognize the standard form of a first-order linear differential equation: $\frac{dy}{dx} + Py = Q$. Here, $P = 2$ and $Q = \sin(2x)$.

The genius of this method lies in the Integrating Factor ($IF$). We are looking for a function that, when multiplied by our entire equation, turns the left side into the derivative of a product. That function is defined as:

$$IF = e^{\int P dx} = e^{\int 2 dx} = e^{2x}$$

This $e^{2x}$ is our magic key; it unlocks the equation by allowing us to rewrite the left side as $\frac{d}{dx}(y \cdot e^{2x})$.

### Tackling the Integral

With our $IF$ in hand, the general solution becomes:

$$y \cdot e^{2x} = \int (\sin(2x) \cdot e^{2x}) dx + C$$

Now, we face the heart of the problem: the integral $\int e^{2x} \sin(2x) dx$. We use the elegant standard formula:

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

With $a = 2$ and $b = 2$, the integral simplifies beautifully to:

$$\frac{e^{2x}}{4} (\sin 2x - \cos 2x)$$

### Finding the Specific Curve

We now have the general solution:

$$y \cdot e^{2x} = \frac{e^{2x}}{4} (\sin 2x - \cos 2x) + C$$

But we need the specific curve that passes through $(0, \frac{3}{4})$. By substituting $x = 0$ and $y = \frac{3}{4}$, we find:

$$\frac{3}{4} \cdot e^0 = \frac{e^0}{4} (\sin 0 - \cos 0) + C$$

Since $e^0 = 1$, $\sin 0 = 0$, and $\cos 0 = 1$, this simplifies to $\frac{3}{4} = -\frac{1}{4} + C$, which gives us **$C = 1$**.

### The Final Evaluation

Our particular solution is:

$$y = \frac{1}{4} (\sin 2x - \cos 2x) + e^{-2x}$$

Finally, we evaluate this at $x = \frac{\pi}{8}$. As we substitute, the term $2x$ becomes $\frac{\pi}{4}$.

We know that $\sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$. Consequently, $(\sin \frac{\pi}{4} - \cos \frac{\pi}{4}) = 0$.

The entire first term vanishes, leaving us with the elegant result:

$$y\left(\frac{\pi}{8}\right) = e^{-2(\frac{\pi}{8})} = **e^{-\frac{\pi}{4}}**$$

Animated Solution for Mathematics - Differential Equations: If $y = y (x)$ is the solution of the differential equation $\frac{d y}{d x} + 2 y = sin (2 x), y (0) = \frac{3}{4}$ , then $y (\frac{π}{8})$ is equal to:

Select Answer:

Visualized Solution

Identify Equation Type

Identify $P$ , $Q$ and $I F$ Formula

Calculate the Integrating Factor

General Solution Setup

The Integral Formula

Evaluate the Integral

The General Solution

Initial Condition

Substitute Initial Condition

Solve for $C$

The Particular Solution

Target Value Setup

Substitute $x = \frac{π}{8}$

Final Answer

The Sigma Insight: Linear Differential Equations

Analyzing the Setup

The Magic Multiplier

Tackling the Integral

Finding the Specific Curve

The Final Evaluation

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