GATE 2022 CH – Question 46
The equation $\frac{dy}{dx} = xy^2 + 2y + x - 4.5$ with the initial condition $y(x = 0) = 1$ is to be solved using a predictor-corrector approach. Use a predictor based on the implicit Euler's method and a corrector based on the trapezoidal rule of integration, each with a full-step size of 0.5. Considering only positive values of $y$, the value of $y$ at $x = 0.5$ is ___________ (rounded off to three decimal places).
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Correct answer: 0.865 to 0.885
Explanation
Predictor (implicit Euler): $y_p = 1 + 0.5(0.5y_p^2 + 2y_p + 0.5 - 4.5)$, that is $y_p = 1 + 0.25y_p^2 + y_p - 2$, so $0.25y_p^2 = 1$ and $y_p = 2$ (positive root). Corrector (trapezoidal rule): $f_0 = 0 + 2 + 0 - 4.5 = -2.5$ and $f(0.5, 2) = 0.5 \times 4 + 4 + 0.5 - 4.5 = 2$. So $y_1 = 1 + 0.25(-2.5 + 2) = 0.875$.