GATE 2020 IN Q.2: solution and answer

GATE 2020 · INQ.2 · 1 mark · MCQ

Consider the recursive equation $X_{n + 1} = X_{n} - h (F (X_{n}) - X_{n})$ , with initial condition $X_{0} = 1$ and $h > 0$ being a very small valued scalar. This recursion numerically solves the ordinary differential equation ____

(A)
$\dot{X} = - F (X), X (0) = 1$
(B)
$\dot{X} = - F (X) + X, X (0) = 1$
(C)
$\dot{X} = F (X), X (0) = 1$
(D)
$\dot{X} = F (X) + X, X (0) = 1$

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

(B) $\dot{X} = - F (X) + X, X (0) = 1$

Official answer: BSource: Official answer key, GATE 2020 IN (IIT Delhi)

Forward (explicit) Euler method: for $\dot{X} = f (X)$ with step size $h$ , the update rule is $X_{n + 1} = X_{n} + h f (X_{n}) .$
Put the given recursion in that form by taking the minus sign inside the bracket: $X_{n + 1} = X_{n} - h (F (X_{n}) - X_{n}) = X_{n} + h (- F (X_{n}) + X_{n}) .$
Matching the two forms term by term gives $f (X) = - F (X) + X$ .
Check by taking the limit: $\frac{X _{n + 1} - X _{n}}{h} = - F (X_{n}) + X_{n}$ . As $h \to 0$ the left side tends to $\dot{X}$ , so $\dot{X} = - F (X) + X$ .
The recursion starts from $X_{0} = 1$ , so the initial condition is $X (0) = 1$ .
The ODE is therefore $\dot{X} = - F (X) + X, X (0) = 1$ , which is option (B).

KEY IDEA

A recursion of the form $X_{n + 1} = X_{n} + h f (X_{n})$ is forward Euler for $\dot{X} = f (X)$ . Rearrange the given update until it has exactly that shape, then read off $f$ . The starting value $X_{0}$ gives the initial condition.

WATCH OUT FOR

Getting the sign wrong on the $X_{n}$ term inside the bracket. $- h (F - X_{n})$ expands to $- h F + h X_{n}$ , so $f = - F + X$ , not $- F - X$ . Students who read only the $- h F (X_{n})$ part pick (A); students who drop the minus sign in front of $h$ pick (C) or (D).

From the GATE 2020 IN paper. Every question of this paper · Download the full paper · All 6 Numerical methods (IN) questions, with notes

GATE 2020 IN Q.2: solution and answer

Answer and solution

More Numerical methods (IN) questions