5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 331

Example 2

Use Euler’s Method with a step size ofΔx= 0 .1 to computey(1) ify(x) is the solution of

the differential equation
dy
dx

• 3 x^2 y= 6 x^2 with initial conditiony(0)=3.

Step 1: For case of the evaluation, transform
dy
dx

• 3 x^2 y= 6 x^2 to
  dy
  dx
  = 3 x^2 (2−y).

Step 2: Create a table showing the iterations. A simple problem, stored in your calculator
and modified with the new differential equation and initial condition, will allow
you to generate the table quickly.

xy 3 x^2 (2−y) yn=yn− 1 +Δx. f′(xn− 1 ,yn− 1 )

03 0 3

0.1 3 − 0. 03 2.997

0.2 2.997 − 0. 11964 2.985036

0.3 2.985036 − 0. 265959 2.9584400

0.4 2.958440 − 0. 460051 2.912434

0.5 2.912434 − 0. 684326 2.844002

0.6 2.844002 − 0. 911522 2.752850

0.7 2.752850 − 1. 106689 2.642181

0.8 2.642181 − 1. 232987 2.518882

0.9 2.518882 − 1. 260884 2.392793

1 2.392793

y(1)≈2.393

Example 3

Use Euler’s Method to approximateP(4), given
dP
dt

=. 3 P

(

1 −

P

20

)

with initial condition

P(0)=4. Use an increment ofΔt= 0 .5.