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464 DIFFERENTIAL EQUATIONS y that x = 0 when y = 1, in the range x = 0(0.2)1.0. [see Table 49.5] 3.0 Table 49.5 x y 2.5 0 1 0.2 1 0.4 0.96 0.6 0.8864 0.8 0.793664 1.0 0.699692 2.0 Figure 49.8 0 0.1 0.2 0.3 0.4 0.5 x Euler’s method of numerical solution of differential equations is simple, but approximate. The method is most useful when the interval h is small. Now try the following exercise. Exercise 185 Further problems on Euler’s method 1. Use Euler’s method to obtain a numerical solution of the differential equation dy dx = 3 − y , with the initial conditions that x x = 1 when y = 2, for the range x = 1.0 to x = 1.5 with intervals of 0.1. Draw the graph of the solution in this range. Table 49.4 x y 1.0 2 1.1 2.1 1.2 2.209091 1.3 2.325000 1.4 2.446154 1.5 2.571429 [see Table 49.4] conditions 2. Obtain a numerical solution of the differential equation 1 dy + 2y = 1, given the initial x dx 3.(a) The differential equation dy dx + 1 =−y x has the initial conditions that y = 1at x = 2. Produce a numerical solution of the differential equation in the range x = 2.0(0.1)2.5. (b) If the solution of the differential equation by an analytical method is given by y = 4 x − x , determine the percentage error 2 at x = 2.2. [(a) see Table 49.6 (b) 1.206%] Table 49.6 x y 2.0 1 2.1 0.85 2.2 0.709524 2.3 0.577273 2.4 0.452174 2.5 0.333334 4. Use Euler’s method to obtain a numerical solution of the differential equation dy dx = x − 2y , given the initial conditions x that y = 1 when x = 2, in the range x = 2.0(0.2)3.0. If the solution of the differential equation is given by y = x2 , determine the percentage 4 error by using Euler’s method when x = 2.8. [see Table 49.7, 1.596%]
NUMERICAL METHODS FOR FIRST ORDER DIFFERENTIAL EQUATIONS 465 Table 49.7 x y 2.0 1 2.2 1.2 2.4 1.421818 2.6 1.664849 2.8 1.928718 3.0 2.213187 49.4 An improved Euler method In Euler’s method of Section 49.2, the gradient (y ′ ) 0 at P (x0 , y 0 ) in Fig. 49.9 across the whole interval h is used to obtain an approximate value of y 1 at point Q. QR in Fig. 49.9 is the resulting error in the result. y z 0 x 0 1 x 0 + h 2 x 1 x Figure 49.10 P h S The following worked problems demonstrate how equations (3) and (4) are used in the Euler-Cauchy method. Q T R y P Q R Problem 4. Apply the Euler-Cauchy method to solve the differential equation dy dx = y − x in the range 0(0.1)0.5, given the initial conditions that at x = 0, y = 2. y 0 0 x 0 x 1 x Figure 49.9 h In an improved Euler method, called the Euler- Cauchy method, the gradient at P (x0 , y 0 ) across half the interval is used and then continues with a line whose gradient approximates to the gradient of the curve at x 1 , shown in Fig. 49.10. Let y P1 be the predicted value at point R using Euler’s method, i.e. length RZ, where y P1 = y 0 + h( y ′ ) 0 (3) The error shown as QT in Fig. 49.10 is now less than the error QR used in the basic Euler method and the calculated results will be of greater accuracy. The corrected value, y C1 in the improved Euler method is given by: y C1 = y 0 + 1 2 h[( y′ ) 0 + f (x 1 , y P1 )] (4) dy dx = y′ = y − x Since the initial conditions are x 0 = 0 and y 0 = 2 then (y ′ ) 0 = 2 − 0 = 2. Interval h = 0.1, hence x 1 = x 0 + h = 0 + 0.1 = 0.1. From equation (3), y P1 = y 0 + h(y ′ ) 0 = 2 + (0.1)(2) = 2.2 From equation (4), y C1 = y 0 + 1 2 h[(y′ ) 0 + f (x 1 , y P1 )] = y 0 + 1 2 h[(y′ ) 0 + (y P1 − x 1 )], in this case = 2 + 2 1 (0.1)[2 + (2.2 − 0.1)] = 2.205 (y ′ ) 1 = y C1 − x 1 = 2.205 − 0.1 = 2.105 If we produce a table of values, as in Euler’s method, we have so far determined lines 1 and 2 of Table 49.8. The results in line 2 are now taken as x 0 , y 0 and (y ′ ) 0 for the next interval and the process is repeated. I
Page 1 and 2: Differential equations 46 Solution
Page 3 and 4: SOLUTION OF FIRST ORDER DIFFERENTIA
Page 5 and 6: α is the temperature coefficient o
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Page 9 and 10: Differential equations 47 Homogeneo
Page 11 and 12: HOMOGENEOUS FIRST ORDER DIFFERENTIA
Page 13 and 14: Differential equations 48 Linear fi
Page 15 and 16: Hence from equation (4): ye x = xe
Page 17 and 18: LINEAR FIRST ORDER DIFFERENTIAL EQU
Page 19 and 20: NUMERICAL METHODS FOR FIRST ORDER D
Page 21: NUMERICAL METHODS FOR FIRST ORDER D
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Page 35 and 36: SECOND ORDER DIFFERENTIAL EQUATIONS
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Page 49 and 50: Differential equations 52 Power ser
Page 51 and 52: POWER SERIES METHODS OF SOLVING ORD
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AN INTRODUCTION TO PARTIAL DIFFEREN
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Differential equations Assignment 1
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