AP Calc- Euler's Method

Question

Use Euler's method with step size 0.1 to estimate y(2.5), where y(x) is the solution of the initial-value problem y ' = 3y + 2xy, y(2) = 1. (Round the answer to four decimal places.) y(2.5)=?

Answer 1

It is really easier to write a simple program, which I will outline here. Or you can do the manual steps until you get to y(2.5).

dy/dx = f(x,y)=3y+2xy x₀ is known = 2, y₀ = 1, h= 0.1

for j = 1 to n // Or how many times you want to repeat to meat your goal

a) m = f(x₀,y₀)

b) y₁ = y₀ + hm

c) t₁ = t₀ + h

d) t₀ = t₁

e) y₀ = y₁

For this problem:

f(x,y) = 3y+2xy

So, starting at t₀ = 2, repeat the above sequence until t₀>2.5

a) m = f(2,1) = 3(1)+2(2)(1) = 7

b) y₁ = y₀ + hm = 1+0.1(7) = 1.7

c) t₁ = t₀ + h = 2.1

d) show t₁ and y₁

e) t₀ = t₁ = 2.1

f) y₀ = y₁ = 1.7

2nd iteration

a) m = f(2.1,1.7) = 3(1.7)+2(2.1)(1.7) = 5.1 + 7.14 = 12.24

b) y₁ = y₀ + hm = 1.7+0.1(12.24) = 1.7 + 1.224 = 2.924

c) t₁ = t₀ + h = 2.2

d) show t₁ and y₁

e) t₀ = t₁ = 2.2

f) y₀ = y₁ = 2.924

---etc---

This is what I got for all iterations from a simple program

x y

2.1 1.7

2.2 2.924

2.3 5.08776

2.4 8.9544576

2.5 15.938934528

y(2.5) = 15.9389