The differential equation dx/dt = (4 - x)/tau, with x(0) = 0, and the constant tau > 0, is to be numerically integrated using the forward Euler method with a constant integration t... The differential equation dx/dt = (4 - x)/tau, with x(0) = 0, and the constant tau > 0, is to be numerically integrated using the forward Euler method with a constant integration time step T. The maximum value of T such that the numerical solution of x converges is: (a) tau/4 (b) tau/2 (c) tau (d) 2tau. Read more

Steps to Solve

1. Identify the Differential Equation

The given differential equation is

$$ \frac{dx}{dt} = \frac{4 - x}{\tau}. $$

We need to apply the forward Euler method for numerical integration.

2. Forward Euler Method Formulation

In the forward Euler method, the next step can be computed using the formula:

$$ x_{n+1} = x_n + T \cdot f(x_n), $$

where $f(x) = \frac{4 - x}{\tau}$ for our equation. Thus, we have:

$$ x_{n+1} = x_n + T \cdot \frac{4 - x_n}{\tau}. $$

3. Stability Condition

For the forward Euler method, we require stability. The condition for stability is typically determined by ensuring that the coefficient of $x_n$ remains less than or equal to 1:

From our equation, we can rewrite:

$$ x_{n+1} = x_n \left(1 - \frac{T}{\tau}\right) + \frac{4T}{\tau}. $$

4. Determine the Stability Bound

Setting the coefficient of $x_n$ for stability gives us:

$$ 1 - \frac{T}{\tau} \leq 1, $$

which simplifies to:

$$ \frac{T}{\tau} < 1 \implies T < \tau. $$

5. Finding Maximum Time Step T

To ensure convergence, we use the more restrictive condition:

$$ T \leq \frac{\tau}{4}. $$

So, the maximum time step $T$ such that the numerical solution converges is given by:

$$ T = \frac{\tau}{4}. $$