Question

The initial value problem \[ \frac{dy}{dt} = f(t, y), t > 0, y(0) = 1, \] where \( f(t, y) = -10 y \), is solved by the following Euler method: \[ y_{n+1} = y_n + h f(t_n, y_n), n \geq 0, \text{with step-size} \, h. \] Then \( y_n \to 0 \) as \( n \to \infty \), provided

• A. \( 0 < h < 0.2 \)
• B. \( 0.3 < h < 0.4 \)
• C. \( 0.4 < h < 0.5 \)
• D. \( 0.5 < h < 0.55 \)

Hint:

For solving differential equations using the Euler method, ensure that the step-size \( h \) is small enough to ensure convergence, especially when solving exponential decay problems.

Answer 1

The Correct Option is A

For the given initial value problem, the exact solution to the differential equation \( \frac{dy}{dt} = -10y \) is: \[ y(t) = e^{-10t}. \] The Euler method provides an approximation to this solution. The general formula for the Euler method is: \[ y_{n+1} = y_n + h f(t_n, y_n). \] Substituting \( f(t, y) = -10y \), we get: \[ y_{n+1} = y_n - 10h y_n. \] This is a simple exponential decay equation. For the approximation to approach zero as \( n \to \infty \), the step size \( h \) must be small enough to avoid significant numerical instability. The condition for stability and convergence of the Euler method is \( 0 < h < 0.2 \), as larger step sizes will lead to growing errors. Final Answer: (A) \( 0 < h < 0.2 \)