Question

An explicit forward Euler method is used to numerically integrate the differential equation

\(\frac{{dy}}{{dt}} = y\)

using a time step of 0.1. With the initial condition y (0) = 1, the value of y (1) computed by this method is ___________ (correct to two decimal places).

Answer 1

Concept:

Euler's Method to generate a numerical solution to an initial value problem of the form:

y'= f (t, y), y(t₀) = y₀

y_n+1 = y_n + h f (t_n, y_n)

Calculation:

\(\frac{{dy}}{{dt}} = y \Rightarrow y' = y\)

t₀ = 0, y₀ = 1, h = 0.1

y_n+1 = y_n + h f (t_n, y_n) = y_n + hy_n

y (0.1) = y₁ = y_o + h y_o = 1 + 0.1 × 1 = 1.1

y (0.2) = y₂ = y₁ + h y₁ = 1.1 + 0.1 × 1.1 = 1.1 (1 + 0.1) = 1.21

y (0.3) = y₃ = y₂ + h y₂ = 1.21 + 0.1 × 1.21 = 1.21 (1 + 0.1) = 1.331

y (0.4) = y₄ = y₃ + h y₃ = 1.331 + 0.1 × 1.331 = 1.331 (1 + 0.1) = 1.4641

y (0.5) = y₅ = y₄ + h y₄ = 1.4641 + 0.1 × 1.4641 = 1.4641 (1 + 0.1) = 1.61051

y (0.6) = y₆ = y₅ + h y₅ = 1.61051 + 0.1 × 1.61051 = 1.61051 (1 + 0.1) = 1.771561

y (0.7) = y₇ = y₆ + h y₆ = 1.771561 + 0.1 × 1.771561 = 1.771561 (1 + 0.1) = 1.9487171

y (0.8) = y₈ = y₇ + h y₇ = 1.9487171 + 0.1 × 1.9487171 = 1.9487171 (1 + 0.1) = 2.14358881

y (0.9) = y₉ = y₈ + h y₈ = 2.14358881 + 0.1 × 2.14358881 = 2.14358881 (1 + 0.1) = 2.357947691

y (1) = y₁₀ = y₉ + h y₉ = 2.357947691 + 0.1 × 2.357947691 = 2.357947691 (1 + 0.1) = 2.59374246

y (1) = 2.59