Question

The number of arbitrary constants in the general solution of the differential equation dy/dx + y = 0 is:

• (A) 0
• (B) 1
• (C) 2
• (D) 3

Answer 1

The Correct Option is B

Step 1: Identify the differential equation
The provided differential equation is dy/dx + y = 0, classified as a first-order linear differential equation.

Step 2: Standard form of a first-order linear differential equation
The general form is dy/dx + P(x)y = Q(x). For this equation, P(x) = 1 and Q(x) = 0.

Step 3: Solution methodology - Integrating Factor
Employ the integrating factor method. The integrating factor (IF) is calculated as e^(∫P(x) dx) = e^(∫1 dx) = e^x.

Step 4: Application of the integrating factor
Multiply the differential equation by the integrating factor: e^x * dy/dx + e^x * y = 0. This simplifies to d/dx (y * e^x) = 0.

Step 5: Integration of both sides
Integrate both sides of the equation: ∫ d/dx (y * e^x) dx = ∫ 0 dx. The result is y * e^x = C, where C represents the constant of integration.

Step 6: Derivation of the general solution
The general solution is y = C * e^(-x).

Step 7: Determination of arbitrary constants
The general solution includes one arbitrary constant, denoted as C.

Final Answer: (B) 1