Consider the differential equation with initial condition

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the differential equation

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

with initial condition

y(0)=1

. A sequence of points

P_0,P_1,P_2,\dots

is generated in the

xy

-plane by the following iterative procedure: beginning at

P_0=(0,1)

, each subsequent point

P_{k+1}

is obtained from

P_k=(x_k,y_k)

by moving horizontally

\Delta x=0.25

unit to the right and vertically an amount equal to the product of

\Delta x

and the slope

\bigl(\tfrac{dy}{dx}\bigr)_{(x_k,y_k)}

. (a) Compute the coordinates of

P_1

P_2

, and

P_3

produced by this procedure. (b) Organize the results from part (a) in a table showing

x_k

y_k

, and the slope used at each step. (c) The exact solution to the differential equation that satisfies

y(0)=1

y=-x-1+2e^{x}

. Determine whether the approximate values

y_1

y_2

, and

y_3

obtained in part (a) lie above or below the exact solution values at the corresponding

x

–coordinates, and provide mathematical justification. (d) Describe how decreasing

\Delta x

0.125

would be expected to affect the accuracy of the approximation at

x=0.75

, and explain your reasoning without performing additional calculations.