1. Euler's method
Rita Korsunsky

2. Using tangent lines Making a chart Calculator
Euler’s method: Approximation of the solution to a differential equation
Using tangent lines
Making a chart
Calculator

3. Given: dy/dx = 1+y and y(0)=0 and ∆x=0
Given: dy/dx = 1+y and y(0)=0 and ∆x=0.5 Construct the approximate y using 3 steps.
First: construct a tangent line at (0,0)
Slope =1
equation y-0 = 1(x-0)
y = x
new y y(0.5) = 0.5
starts at (0,0)
ends at (0.5,0.5)
0.5

4. Given: dy/dx = 1+y and y(0)=0 and ∆x=0.5
Second: construct a tangent line at (.5,.5)
slope = 1.5
equation y - .5 = 1.5(x-.5)
y = 1.5x-.25
new y y(1) = =1.25
starts at (.5,.5)
ends at (1,1.25)
2.5
2.0
1.5
1.0
0.5

5. Given: dy/dx = 1+y and y(0)=0 and ∆x=0.5
Third: construct a tangent line at (1,1.25)
slope = 2.25
equation y = 2.25(x-1)
y = 2.25x-1
new y y(1.5) = = 2.375
starts at (1,1.25)
ends at (1.5,2.375)
2.0
1.5
1.0
0.5

6. Find the exact solution by
solving the differential equation:

7. Compare the exact y and approximate y
(1.5,2.375)
y = ex - 1
approximate y
(1,1.25)
(0.5,0.5)
link

8. Given: dy/dx =2x ,y(1)=3. Find y(2) using 5 steps.
Basic Idea: new y ≈ old y + dy/dx (∆x)
Step size =∆x=(2-1)/5 = 1/5 x
old y
new y ≈ old y +
dy/dx (∆x) 1
1.2
1.4
1.6
1.8
2.0 3
2(1/5)=0.40
3+0.40=3.40
3.4
2.4(1/5)=0.48
=3.88
3.88
2.8(1/5)=0.56
=4.44
4.44
3.2(1/5)=0.64
=5.08
5.08
3.6(1/5)=0.72
=5.80
5.80

9. Euler’s method on calculator
Given: dy/dx =2x ,y(1)=3. ∆x= 1/5. Find y(2).
MODE: Change Graph Mode to Diff. Equations
Press and F1 (Y=) Enter initial x condition under t0= and initial y under yi1=
Type in the diff. equation for y1. Use t for x and y1 for y when inputting the differential equation 1
1.2
1.4
1.6
1.8
2.0
3.40
3.88
4.44
5.08
5.80 3
While still on the Y= page, press F1 and scroll down to 9: Format, press enter
Scroll down to Solution Method and choose Euler
Press F2 to go to the Window Page. Change the tstep to your
Press and F4 (TblSet) and input your initial x under tblStart and the under
and F5 to view the table

10. Euler’s Example Given: Solution: On the Y= Menu, When x=2.5, y=-5 t y14
1.5 -2 2
2.5 -5
When x=2.5, y=-5