Solve and select the correct answer.

Explanation: Method 1: Variable Separation Step 1: Equation ko rearrange karna Humein diya gaya hai: Ise hum aise likh sakte hain: Step 2: Variables ko alag karna waale terms ko ke saath aur ko doosri taraf le jaate hain: Step 3: Integrate karna Dono taraf integration karne par: Humein pata hai ki , isliye: Step 4: ke liye solve karna Dono side ko se multiply karte hain: Ab exponential ( ) lete hain: Maan lijiye (ek naya constant), toh: Method 2: Integrating Factor (Linear Form) Ye equation ki form mein hai, jahan aur . Step 1: Integrating Factor ( ) nikaalna Step 2: Solution ka formula lagana Step 3: se divide karna

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MAH-CET 2025 — Mathematics PYQ

MAH-CET | Mathematics | 2025

Solve $\frac{d y}{d x} + y = 1$ and select the correct answer.

Choose the correct answer:

(Correct Answer)

B.
$y = C e^{x} - 1$

C.
$y = C e^{- x}$

D.
$y = 1 - C e^{x}$

Explanation

Method 1: Variable Separation

Step 1: Equation ko rearrange karna

Humein diya gaya hai:

$\frac{dy}{dx} + y = 1$

Ise hum aise likh sakte hain:

$\frac{dy}{dx} = 1 - y$

Step 2: Variables ko alag karna

$y$ waale terms ko $dy$ ke saath aur $dx$ ko doosri taraf le jaate hain:

$\frac{dy}{1 - y} = dx$

Step 3: Integrate karna

Dono taraf integration karne par:

$\int \frac{dy}{1 - y} = \int dx$

Humein pata hai ki $\int \frac{1}{a-x} dx = -\ln|a-x|$ , isliye:

$-\ln(1 - y) = x + C_1$

Step 4: $y$ ke liye solve karna

Dono side ko $-1$ se multiply karte hain:

$\ln(1 - y) = -x - C_1$

Ab exponential ( $e$ ) lete hain:

$1 - y = e^{-x - C_1}$

$1 - y = e^{-x} \cdot e^{-C_1}$

Maan lijiye $e^{-C_1} = C$ (ek naya constant), toh:

$1 - y = Ce^{-x}$

$y = 1 - Ce^{-x}$

Method 2: Integrating Factor (Linear Form)

Ye equation $\frac{dy}{dx} + Py = Q$ ki form mein hai, jahan $P = 1$ aur $Q = 1$ .

Step 1: Integrating Factor ( $IF$ ) nikaalna

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

Step 2: Solution ka formula lagana

$y \cdot (IF) = \int (Q \cdot IF) dx + C$

$y \cdot e^x = \int (1 \cdot e^x) dx + C$

$y \cdot e^x = e^x + C$

Step 3: $e^x$ se divide karna

$y = \frac{e^x}{e^x} + \frac{C}{e^x}$

$y = 1 + Ce^{-x}$