Mathematics

Problem

The function

y=y(x)

satisfies the differential equation

\frac{dy}{dx}=x^{2}-y

with initial condition

y(0)=0

. Approximations are to be generated for

y(0.5)

y(1.0)

, and

y(1.5)

using Euler’s method with constant step size

h=0.5

. (a) Show that

\dfrac{d^2y}{dx^2}=2x-\dfrac{dy}{dx}

. (No solution of the differential equation is required.) (b) Using the differential equation and your result from part (a), determine all

x

in the interval

0\le x\le1.5

at which the solution curve changes concavity. Explain your reasoning. (c) Carry out Euler’s method to compute approximations to

y(0.5)

y(1.0)

, and

y(1.5)

. Present your values in an organized table. (d) For each of the three approximations obtained in part (c), state whether it is greater than or less than the corresponding exact value of

y

. Provide full justification using the concavity information from part (b) together with the geometric relationship between the tangent line and the curve.

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