GATE 2020 IN – Question 12

Engineering Mathematics · Numerical methods: Solution of matrix equations, non-linear algebraic equations, iterative methods for differential equations, numerical integration, regression and correlation analysis, backpropagation for neural networks · 1 mark · Multiple choice

The recursion is $X_{n+1}=X_n-h[F(X_n)-X_n]$, with $X_0=1$ and small positive h. It numerically solves which ODE?

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

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Correct answer: (B) $\dot X=-F(X)+X,\ X(0)=1$

Explanation

Forward Euler for $\dot X=g(X)$ is $X_{n+1}=X_n+hg(X_n)$. Comparing this with the given step gives $g(X)=-[F(X)-X]=-F(X)+X$. The initial value is directly $X(0)=X_0=1$. Thus B matches both the derivative and initial condition.

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