Mathematics

Problem

Consider the initial–value problem

\frac{dy}{dx}=2x-y,\qquad y(0)=1.

You will approximate

y(0.4)

using Euler’s method with step size

h=0.2

. (a) Complete the Euler table for

x=0

0.2

, and

0.4

, and record the resulting approximation for

y(0.4)

. (b) Derive a formula for

\dfrac{d^2y}{dx^2}

expressed only in terms of

x

and

y

. Determine the sign of

\dfrac{d^2y}{dx^2}

along the solution curve for

0\le x\le0.4

. Justify your answer without solving the differential equation. (c) Based on the concavity established in part (b), decide whether the Euler approximation from part (a) is an overestimate or an underestimate of the exact value of

y(0.4)

. Explain thoroughly.

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