Mathematics

Problem

The differential equation

\dfrac{dy}{dx}=x+\dfrac{y}{2}

models a quantity

y

that changes with respect to

x

. The initial condition

y(0)=1

is given. (a) Draw the direction field for the differential equation on the grid

-0.5\le x\le1

and

0\le y\le3

, using a lattice spacing of

0.25

in both directions. (b) Sketch an approximate solution curve that satisfies the initial condition

y(0)=1

and extends to

x=1

. (c) Apply Euler’s method with step size

\Delta x=0.5

to approximate

y(1)

. Present your work in a table that includes the slope used at each step and the resulting estimate of

y

. (d) Compare your Euler approximation from part (c) with the value of

y(1)

suggested by your sketch in part (b). Provide a reasoned explanation for any discrepancy, referring to features of the direction field.