Eulers mehod. Leonard Euler (1707 1783) Leonhard Euler was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics like calculus and differential equations. dy f ( x, yusing ) Numerical solution of Eulers method. dx yn 1 yn h f xn , yn xn 1 xn h, where h is a constant (step length) The above formulas are given in the formulae booklet so no need to memorise.
Eulers Method. Allows to find an approximate solution to a differentialdyequation f ( x, y ) dx This is done by finding a y0 ,points sequence x0 ,of x1 , y1 , x2 , y2 ..... xn , yn which lie on the solution curve of the given differential equation with a boundary condition. We need to approximate numerically the value of yn at xn. As numerical calculations are quite time consuming when performed manually, the spreadsheet or computer programs are helpful to do all the calculations much quicker. Graphically we start at the point
( 0 , 0) and find the gradient at this point. The step h is usually given in the question. The smaller the step, the more accurate the ( 1, 1) approximation. We move to the point by using the gradient calculated in step 1. ( , ) We repeat the process until we reach the point yn1 yn h f xn , yn xn 1 xn h Iterations, how it works: y1 y0 h f x0 , y0 x1 x0 h y2 y1 h f x1 , y1
x2 x1 h and so on Example 1: Use Eulers method of numerical integration with a step size of 0.1 to find y(1) if y=y-x and y(0)=2. Identify the information given: f ( x, y ) y x h 0.1 x0 0, y0 2 Now multiply by h 2 0.1 0.2 First calculate f(0,2) f (0, 2) y0 x0 2 0 2 Add to y0
0.2 2 2.2 y1 yn1 yn h f xn , yn xn 1 xn h Repeat the process with the point (x1,y1) x1 0.1, y1 2.2 To obtain the point (x2,y2) x1 0.2, y1 2.41 Repeating the process eight more times and tabulating the results gives the following: n xn 0
0 1 yn f(x,y)h yn+1 = yn +f(x, y) h 2 0.2 2.2 0.1 2.2 0.21 2.41
4.258 9 0.8 4.258 0.3358 4.59374 10 1.0 4.59374 How accurate is this result? When solving the above DE we obtain y=ex+x+1 when x 1, yexact e 2 y 4.71828
So we have an underestimation here 4.59374 4.71828 = 0.12454 and the absolute percentage error of around 2.64%, so not a bad approximation. This value can be also obtained using your TI Nspire, sketch the slope field in the Diff Eq mode, enter the initial condition. Use Trace, Graph trace and left-right arrows to move along the curve in steps of 0.1 (default) Edit parameters to change the step. Make sure Euler is Using a spreadsheet on TI Nspire (Lists & Spreadsheet). Steps: Name the columns. Fill in the values for x from 0 to 1 every 0.1. Enter the initial value for y as 2. In cell c1=0.1(b1-a1) In cell d1=b1+c1 In cell b2=d1 And copy down one by one.
Using a Calculator RHS of DE initial x, final x variables y-value from initial condition step answer Your dy sin x turn ! eDE Use Eulers method to find y(0.5) for the y(0)=1 and h=0.1 dx
with What to expect on the examination? dy x y 2 and dx Consider the differential equation y (1) 0. Use Eulers method with steps of 0.2 to estimate f(2) to 5 decimal places. f(2)=1.10033 Note that the step is 0.2 now and required accuracy is 5 d.p. Change the float in 9: Settings Display Digits Float to display 5 d.p.
dy x Question: 2 y with y(0)=3, use the Eulers method (a) For the differential equationdx with step h=0.5 to find y(7) to 4 decimal places. (b)Sketch the slope field for the above differential equation for x values from 0 to 7 and y values from -4 to 4. (c) Plot the pairs of x and y-values from your spreadsheet on the slope field graph. (d)For which values of x does the numerical solution by Eulers method seem to follow the slope field? (e) For which values of x is the numerical method clearly wrong? (f) Solve the differential equation algebraically. (g)Plot the particular solution which contains (0,3). (h)Explain why Eulers method gives meaningless answers for larger values Answers: of x. (a) f(7)=2.0634 (d) Eulers method seems to produce reasonable answers for x<5 (f) 0.5x2+y2=9 (h) The solution curve is an ellipse with semi-major axis = 4.24 (x-direction)
and semi-minor axis =3 (y-direction). When the curve approaches the xaxis, the slope becomes nearly infinite. Furthermore, after crossing the xaxis the curve should go in the negative x-direction, but the method does not account for this. Summary: Describe Eulers method for solving the initial value problem y=f(x,y), y(x0)=y0. Comment on the methods accuracy. Why might you want to solve an initial value problem numerically? What is the slope field of a differential equation y=f(x,y)? What can we learn from such fields?
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