Question

The solution of the differential equation \( 5y\,dx = 2x\,dy \) passing through the point \( (1,1) \) is:

• A. \( 2\ln x = 5\ln y \)
• B. \( 5\ln x = 2\ln y \)
• C. \( \ln(y + x) = 2 \)
• D. \( \ln(1 + xy) = 0 \)
• E. \( 3\ln x = 5\ln y \)

Hint:

Whenever the equation contains both \(x\) and \(y\), first check whether variables can be separated. Most exam questions from differential equations use this standard technique.

Answer 1

The Correct Option is B

Concept: A separable differential equation can be solved by separating variables: \[ \frac{dy}{dx} = f(x)g(y) \] Then integrate both sides separately.

Step 1: Rewrite the differential equation. Given: \[ 5y\,dx = 2x\,dy \] Rearranging: \[ \frac{dy}{dx} = \frac{5y}{2x} \] Now separate variables: \[ \frac{dy}{y} = \frac{5}{2}\frac{dx}{x} \]

Step 2: Integrate both sides. \[ \int \frac{dy}{y} = \int \frac{5}{2}\frac{dx}{x} \] \[ \ln y = \frac{5}{2}\ln x + C \] Multiply by 2: \[ 2\ln y = 5\ln x + C_1 \]

Step 3: Use the point \( (1,1) \). Given: \[ x = 1, y = 1 \] Substitute: \[ 2\ln 1 = 5\ln 1 + C_1 \] Since \[ \ln 1 = 0 \] we get: \[ C_1 = 0 \] Thus, \[ 2\ln y = 5\ln x \] or \[ \boxed{5\ln x = 2\ln y} \]