Let y=y(x) be the solution curve of the differential equation (1+sin x) dy/dx + (y+1) cos x = 0, y(0)=0. If the curve y=y(x) passes through the point (alpha, -1/2), then a value of alpha is :

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q20 – JEE-MAIN 2026

Let

y = y (x)

be the solution curve of the differential equation

(1 + \sin x) \frac{d y}{d x} + (y + 1) \cos x = 0, y (0) = 0

. If the curve

y = y (x)

passes through the point

(α, \frac{- 1}{2})

, then a value of

α

is :

Identify that the given differential equation is a first-order separable differential equation.

Step 1: Separate Variables and Integrate✦ Active

Rearrange the given differential equation to separate variables and integrate both sides:

(1 + \sin x) \frac{d y}{d x} = - (y + 1) \cos x

\frac{d y}{y + 1} = - \frac{\cos x}{1 + \sin x} d x

Integrating both sides yields:

\int \frac{d y}{y + 1} = - \int \frac{\cos x}{1 + \sin x} d x

\ln | y + 1 | = - \ln | 1 + \sin x | + C

\ln | (y + 1) (1 + \sin x) | = C

(y + 1) (1 + \sin x) = A

, where

A = e^{C}

is an arbitrary positive constant.

Step 2: Apply Initial Condition○ Expand

Use the initial condition $y (0) = 0$ to find the value of the constant $A$ .

Substitute

x = 0

and

y = 0

into the general solution:

(0 + 1) (1 + \sin 0) = A

(1) (1 + 0) = A ⟹ A = 1

The particular solution is

(y + 1) (1 + \sin x) = 1

Step 3: Find

α

using the given point○ Expand

The curve passes through the point $(α, - \frac{1}{2})$ . Substitute these values into the particular solution.

(- \frac{1}{2} + 1) (1 + \sin α) = 1

(\frac{1}{2}) (1 + \sin α) = 1

1 + \sin α = 2

\sin α = 1

From the given options,

α = \frac{π}{2}

satisfies

\sin α = 1

💡 Teacher's Secret Hint

Ensure to check all options if multiple values could satisfy the condition, but in this case, only one option matches.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.