Euler's Method

Question

Use Euler's Method with three equal step sizes to estimate the value of y(0.3) for the differential equation y ′ = y, with y(0) = 1.

Answer 1

SOLUTION

Given y(0)=1, n=3 

Step size is h=[(0.3)−(0)]/3 = 0.1

The Euler's method states that yn+1=yn+h⋅f(tn,yn), where tn+1=tn+h.

We have that h=0.1, t0=0, y0=1, f(t,y)=y

 

Step 1.

t1=t0+h=0+0.1=0.1

 

y(t1)=y(0.1)=y1=y0+h⋅f(t0,y0)=1+h⋅f(0,1)=
=1+0.1⋅*(1.0)=1.1

 

Step 2.

t2=t1+h=0.1+0.1=0.2

y(t2)=y(0.2)=y2=y1+h⋅f(t1,y1)=1.1+h⋅f(0.1,1.1)=

=1.1+0.1⋅(1.1)=1.21

Step 3.

t3=t2+h=0.2+0.1= 0.3
y(t3)=y(0.3)=y3=y2+h⋅f(t2,y2)=1.21+h⋅f(0.2,1.21)=

 

=1.21+0.1⋅(1.21)=1.331

Answer: y(310)=1.331