Question

The solution of the differential equation \( x + y \frac{dy}{dx} = 0 \), given that at \( x = 0 \), \( y = 5 \), is:

• A. \( x^2 + y^2 = 5y \)
• B. \( x^2 + 5y^2 = 125 \)
• C. \( x^2 + y = 5 \)
• D. \( x^2 + y^2 = 25 \)
• E. \( 2x^2 + y^2 = 25 \)

Hint:

When solving a first-order linear differential equation, always separate the variables and integrate both sides. Apply initial conditions carefully to determine the constant of integration.

Answer 1

The Correct Option is D

Step 1: Given the differential equation: \[ x + y \frac{dy}{dx} = 0, \] rearrange to separate variables: \[ y \, dy = -x \, dx. \] Step 2: Integrate both sides: \[ \int y \, dy = \int -x \, dx. \] Step 3: Perform the integration: \[ \frac{y^2}{2} = -\frac{x^2}{2} + C, \] where \( C \) is the constant of integration. 
Step 4: Multiply through by 2 to simplify: \[ y^2 = -x^2 + 2C. \] 
Step 5: Use the initial condition \( y = 5 \) when \( x = 0 \) to find \( C \): \[ 5^2 = -0^2 + 2C \quad \Rightarrow \quad 25 = 2C \quad \Rightarrow \quad C = \frac{25}{2}. \] 
Step 6: Substitute \( C \) into the equation: \[ y^2 = -x^2 + 25. \] Thus, the solution to the differential equation is: \[ x^2 + y^2 = 25. \]