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: Understand the differential equation
The given differential equation is dy/dx + y = 0, which is a first-order linear differential equation.

Step 2: General form of a first-order linear differential equation
A first-order linear differential equation can be written as dy/dx + P(x)y = Q(x). In this case, P(x) = 1 and Q(x) = 0.

Step 3: Solve the differential equation
To solve, use the integrating factor method.
The integrating factor (IF) = e^(∫P(x) dx) = e^(∫1 dx) = e^x.

Step 4: Multiply the entire equation by the integrating factor
e^x * dy/dx + e^x * y = 0
This can be written as d/dx (y * e^x) = 0.

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

Step 6: Express the general solution
y = C * e^(-x).

Step 7: Number of arbitrary constants
The general solution contains one arbitrary constant C.

Final Answer: (B) 1