GATE 2025 CE (CE1) – Question 49
Consider the differential equation given below. Using the Euler method with the step size ($h$) of 0.5, the value of $y$ at $x = 1.0$ is equal to ____________ (*rounded off to 1 decimal place*).
$$\frac{dy}{dx} = y + 2x - x^2; \qquad y(0) = 1 \qquad (0 \le x < \infty)$$
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Correct answer: 2.6 to 2.7
Explanation
Euler's method uses $y_{n+1} = y_n + h\,f(x_n, y_n)$. At $x = 0$: $f = 1 + 0 - 0 = 1$, so $y(0.5) = 1 + 0.5 \times 1 = 1.5$. At $x = 0.5$: $f = 1.5 + 1 - 0.25 = 2.25$, so $y(1.0) = 1.5 + 0.5 \times 2.25 = 2.625$, which is about 2.6 to 2.7 to one decimal place.