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QB_EM_ADV_084
Question Bank • QB_EM_ADV_084
NAT • 2 Marks
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Numerical solutions of ODEs
Question Statement:
Consider the initial value problem $\frac{dy}{dx} = x + y^2$ with $y(0) = 1$. Using the modified Euler method (Heun's predictor-corrector method) with step size $h = 0.1$, the value of $y(0.1)$ is ________ (round off to three decimal places).
Step-by-Step Solution & Explanation
Given: $x_0 = 0, y_0 = 1, h = 0.1$, and $f(x, y) = x + y^2$.
**Predictor Step (Forward Euler):**
$k_1 = f(x_0, y_0) = 0 + 1^2 = 1.0$
$y_1^{(p)} = y_0 + h k_1 = 1 + (0.1)(1.0) = 1.10$
**Corrector Step (Average Slope):**
$x_1 = x_0 + h = 0.1$
$k_2 = f(x_1, y_1^{(p)}) = 0.1 + (1.10)^2 = 0.1 + 1.21 = 1.31$
Average slope:
$\bar{k} = \frac{k_1 + k_2}{2} = \frac{1.0 + 1.31}{2} = \frac{2.31}{2} = 1.155$
$y_1 = y_0 + h \bar{k} = 1 + (0.1)(1.155) = 1 + 0.1155 = 1.1155 \approx 1.116$
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