Mathematics

Problem

Consider the differential equation

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

with solution curve that passes through the point

(0,1)

. A student wishes to approximate the value of

y(0.1)

. (a) Evaluate

\dfrac{dy}{dx}

(0,1)

. (b) Compute an explicit expression for

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

in terms of

x

and

y

. Then evaluate

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

(0,1)

and state its sign. (c) The student uses a single tangent‐line step of size

h=0.1

beginning at

(0,1)

. Without carrying out any numerical calculation, determine whether this approximation of

y(0.1)

is greater than or less than the true value. Justify your answer using the concavity you found in part (b).