If y = y(x) and (2 + sin x)/(y+1) (dy/dx) = -cos x, y(0) = 1, then y(π/2) equals
(a) 1/3
(b) 2/3
(c)

Q: Wait, why do we reject the negative root or absolute value signs in $\ln(y+1)$?

That is a brilliant observation! Technically, the integral of $\frac{1}{y+1}$ is $\ln|y+1|$. However, we look at our initial condition: $y(0) = 1$. This means at $x=0$, $y+1 = 2$, which is positive. Since the solution curve is continuous, $y+1$ will remain positive in the neighborhood of our point of interest. Thus, we can safely drop the absolute value bars.

Q: Why can't we just use energy conservation or some other physics principle here?

While physics problems often involve differential equations, this is a pure mathematical exercise in calculus. We are solving for a function $y(x)$ that satisfies a specific rate of change relationship. Stick to the calculus toolkit—separation of variables is your best friend here.

Q: How did $C$ turn into $\ln 4$? I got lost in the log algebra.

Don't worry, log algebra is where many students stumble! We had $\ln(2) = -\ln(2) + C$. Adding $\ln(2)$ to both sides gives $C = 2\ln(2)$. Using the power rule $n\ln(a) = \ln(a^n)$, we get $C = \ln(2^2) = \ln(4)$. It is all about keeping your logarithmic properties sharp!

Question

If $y = y(x)$ and $\frac{2 + \sin x}{y+1} (\frac{dy}{dx}) = -\cos x, y(0) = 1$, then $y(\frac{\pi}{2})$ equals

Options:

(A) 1/3

(B) 2/3

(C) -1/3

(D) 1

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer. Today, we are not just solving a problem; we are uncovering the hidden geometry of a function. We are given the differential equation $\frac{2 + \sin x}{y+1} (\frac{dy}{dx}) = -\cos x$, with the initial condition $y(0) = 1$.

It might look like a jumble of symbols, but I want you to see it as a map. We are looking for a specific path—a curve—that starts at the point $(0, 1)$ and evolves according to the rules defined by this equation.

### The Art of Separation

The first step in any differential equation is to bring order to chaos. We have $x$ terms and $y$ terms tangled together. Our goal is to isolate them.

By multiplying both sides by $dx$ and rearranging the terms, we arrive at a beautiful, clean separation:

$$\frac{1}{y+1} dy = \frac{-\cos x}{2 + \sin x} dx$$

The variables are now perfectly separated. This is the moment where the problem shifts from a differential equation to a simple integration task. We are ready to bridge the gap between the rate of change and the function itself.

### The Calculus Domain

Now, we integrate both sides. On the left, we have the integral of $\frac{1}{y+1}$ with respect to $y$, which yields $\ln(y+1)$.

On the right, we face the integral $\int \frac{-\cos x}{2 + \sin x} dx$. Do you see the elegance? The derivative of the denominator, $2 + \sin x$, is exactly $\cos x$.

We use the substitution $u = 2 + \sin x$, which implies $du = \cos x dx$. The integral transforms into $-\int \frac{1}{u} du$, which is simply $-\ln(u)$. Substituting back, we get our general solution:

$$\ln(y+1) = -\ln(2 + \sin x) + C$$

### Finding the Specific Path

We have a family of curves, but we need the one that passes through $(0, 1)$. We substitute $x=0$ and $y=1$ into our general solution:

$$\ln(1+1) = -\ln(2 + \sin 0) + C$$

Since $\sin(0) = 0$, this simplifies to $\ln(2) = -\ln(2) + C$. Solving for $C$, we find $C = 2\ln(2)$.

Using the power rule of logarithms, $C = \ln(2^2) = \ln(4)$. Our particular solution is now locked in:

$$\ln(y+1) = -\ln(2 + \sin x) + \ln(4)$$

### The Final Reveal

We are in the home stretch. Let's clean up this equation using the quotient rule of logarithms, $\ln A - \ln B = \ln(\frac{A}{B})$:

$$\ln(y+1) = \ln\left(\frac{4}{2 + \sin x}\right)$$

Exponentiating both sides, we get the explicit function:

$$y+1 = \frac{4}{2 + \sin x}$$

Finally, we evaluate this at $x = \frac{\pi}{2}$. Since $\sin(\frac{\pi}{2}) = 1$, the equation becomes:

$$y+1 = \frac{4}{2 + 1} = \frac{4}{3}$$

Subtracting $1$ from both sides, we find **$y = \frac{4}{3} - 1 = \frac{1}{3}$**. We have arrived at the destination. The value is **$\frac{1}{3}$**, which corresponds to option (A).

Animated Solution for Mathematics - Differential Equations: If $y = y (x)$ and $\frac{2 + s i n x}{y + 1} (\frac{d y}{d x}) = - cos x, y (0) = 1$ , then $y (\frac{π}{2})$ equals

Select Answer:

Visualized Solution

Analyze the Differential Equation

Separating the Variables

Setting up the Integration

Integrating the Left-Hand Side

Integrating the Right-Hand Side

The General Solution

Applying the Initial Condition

Solving for the Constant $C$

Simplifying the Particular Solution

Evaluating at $x = \frac{π}{2}$

The Final Answer

The Sigma Insight: Variable Separable Method

Analyzing the Setup

The Art of Separation

The Calculus Domain

Finding the Specific Path

The Final Reveal

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Animated Solution for Mathematics - Differential Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If y=y(x) and y+12+sinx​(dxdy​)=−cosx,y(0)=1, then y(2π​) equals