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).
Concept:
Euler's Method to generate a numerical solution to an initial value problem of the form:
y'= f (t, y), y(t0) = y0
yn+1 = yn + h f (tn, yn)
Calculation:
\(\frac{{dy}}{{dt}} = y \Rightarrow y' = y\)
t0 = 0, y0 = 1, h = 0.1
yn+1 = yn + h f (tn, yn) = yn + hyn
y (0.1) = y1 = yo + h yo = 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
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