Question

What is the solution to the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) with the initial condition \( y(1) = 2 \)?

• A. \( y = 2x \)
• B. \( y = x^2 \)
• C. \( y = 2x^2 \)
• D. \( y = x \)

Hint:

For separable differential equations, integrate both sides after separating the variables to find the general solution.

Answer 1

The Correct Option is A

The differential equation is given as: \[\frac{dy}{dx} = \frac{y}{x}.\] This is a separable differential equation, rewritten as: \[\frac{dy}{y} = \frac{dx}{x}.\] Integrating both sides yields: \[\int \frac{1}{y} dy = \int \frac{1}{x} dx,\] which results in \[\ln |y| = \ln |x| + C.\] Exponentiating both sides gives: \[|y| = e^{\ln |x| + C} = |x| e^C.\] Consequently, \( y = Cx \). Applying the initial condition \( y(1) = 2 \): \[\begin{aligned} 2 &= C(1) \\ C &= 2 \end{aligned}.\] The solution is thus \( y = 2x \).