Let the solution y = y(x) of the differential equation dy/dx - y = 1 + 4 sin x satisfy y(π) = 1. The

Q: Wait, why is $P = -1$ and not $1$? I thought the standard form was $\frac{dy}{dx} + Py = Q$.

That is a fantastic observation! You are absolutely right about the standard form. The equation is $\frac{dy}{dx} - y = 1 + 4 \sin x$. If we rewrite this as $\frac{dy}{dx} + (-1)y = 1 + 4 \sin x$, we can clearly see that $P = -1$. Always be careful with the sign of the coefficient of $y$!

Q: Do I really have to use integration by parts twice for $\int e^{-x} \sin x dx$? That seems so long.

You definitely don't have to! While integration by parts works, it is prone to sign errors. In JEE, we love efficiency. Use the standard formula: $\int e^{ax} \sin bx dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx)$. With $a = -1$ and $b = 1$, you will get the result in one step!

Q: Why does the constant $C$ become zero? Did I make a mistake?

Not at all! The constant $C$ is determined by the initial condition $y(\pi) = 1$. When you substitute $x = \pi$ and $y = 1$ into your general solution, the terms involving $\sin \pi$ and $\cos \pi$ simplify, and you will find that $C$ must be zero to satisfy the condition. It is perfectly normal for $C$ to be zero.

Question

Let the solution $y = y(x)$ of the differential equation $\frac{dy}{dx} - y = 1 + 4 \sin x$ satisfy $y(\pi) = 1$. Then $y(\frac{\pi}{2}) + 10$ is equal to ______

Sigma Percentile · Accepted Answer

### The Beauty of Differential Equations

Differential equations are the language of change. They describe how things evolve, from the cooling of a cup of coffee to the orbits of planets.

Today, we are going to unravel a beautiful problem that tests your ability to recognize patterns and apply standard tools with precision. Let us dive into the equation:

$$ \frac{dy}{dx} - y = 1 + 4 \sin x $$

### Phase 1: Pattern Recognition

Before we rush into calculations, let us pause and look at the structure. This is a first-order linear differential equation.

Its standard form is $\frac{dy}{dx} + Py = Q$. By comparing our equation to this, we identify $P = -1$ and $Q = 1 + 4 \sin x$.

This identification is the foundation of our entire journey. If we get this wrong, everything else will crumble, so take a moment to ensure you have captured the signs correctly.

### Phase 2: The Integrating Factor

Now, we need a bridge to connect the derivative to the function. This bridge is the Integrating Factor ($IF$).

The formula is $IF = e^{\int P dx}$. Substituting our $P = -1$, we get:

$$ IF = e^{\int -1 dx} = e^{-x} $$

This $e^{-x}$ is the magic key that will transform our equation into something we can integrate. It acts as a scaling factor, allowing us to write the left side of our equation as the derivative of a product.

### Phase 3: The Integration Challenge

The general solution is $y \cdot IF = \int (Q \cdot IF) dx + C$. Substituting our values, we get:

$$ y e^{-x} = \int e^{-x}(1 + 4 \sin x) dx + C $$

We can split this into two integrals: $\int e^{-x} dx$ and $4 \int e^{-x} \sin x dx$. The first is simple: $\int e^{-x} dx = -e^{-x}$.

The second, $\int e^{-x} \sin x dx$, is a classic JEE integral. Instead of doing integration by parts twice, we use the standard result:

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

With $a = -1$ and $b = 1$, this becomes:

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

### Phase 4: The Initial Condition

Combining these, we get $y e^{-x} = -e^{-x} - 2 e^{-x} (\sin x + \cos x) + C$. Multiplying by $e^x$ gives us the general solution:

$$ y = -1 - 2(\sin x + \cos x) + C e^x $$

Now, we use the initial condition $y(\pi) = 1$. Substituting $x = \pi$ and $y = 1$, we find:

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

This forces $C = 0$. Our particular solution is $y(x) = -1 - 2(\sin x + \cos x)$.

### Phase 5: The Final Evaluation

Finally, the question asks for $y(\frac{\pi}{2}) + 10$. Substituting $x = \frac{\pi}{2}$ into our particular solution, we get:

$$ y\left(\frac{\pi}{2}\right) = -1 - 2\left(\sin \frac{\pi}{2} + \cos \frac{\pi}{2}\right) = -1 - 2(1 + 0) = -3 $$

Adding $10$ to this result, we get $-3 + 10 = 7$. We have navigated the complexity and arrived at the answer: **7**.

Animated Solution for Mathematics - Differential Equations: Let the solution $y = y (x)$ of the differential equation $\frac{d y}{d x} - y = 1 + 4 sin x$ satisfy $y (π) = 1$ . Then $y (\frac{π}{2}) + 10$ is equal to ______

Enter Numerical Value:

Visualized Solution

Identify the Equation Type

Extract $P$ and $Q$

The Integrating Factor ( $I F$ )

General Solution Setup

Splitting the Integral

The Special Integral Formula

Applying the Formula

Integrating the Constant Term

Combining the Results

Solving for $y (x)$

Apply Initial Condition

Solving for $C$

The Particular Solution

Evaluate $y (\frac{π}{2})$

Final Calculation

The Sigma Insight: Linear Differential Equations

The Beauty of Differential Equations

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

Similar Questions

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