Question

8.2 (2). Consider the linear equation Y'(x) = XY (2) + (1 - 1) cos(x) - (1+) sin(x), quady (0) = 1 The true solution is Y (1) = sin(1) + cos(r). Solve this problem using Euler's method with several values of and h, for 0 <<<10. Comment on the results. (a) X = -1; h=0.5, 0.25, 0.125 (b) X = 1; h = 0.5, 0.25, 0.125 (c) = -5; h=0.5, 0.25, 0.125, 0.0625 (d) = 5; h= 0.0625

Answer 1

Part(a)

1 Your Input: find y (1) for y --y + 2 cos (2), when y (0) = 1, h = using the Euler's method. 2 The Euler's method states that Yn+1 = yn + hºf (<n» Yn). where xn+1 = xm + h. 1 We have that h = 2 20 = 0,40 = 1, f(x, y) = = -y + 2 cos (2) Step 1. 1 1 X1 = x + h = 0 + = 2 2 = y y (21) (9) = y1 = 40 +hºs (2oyo) = 1 +hºf (0,1) = = 1+2 (1:0) – 1.5 Step 2. 1 X2 = X1 + h = 1 - + = 2 2 = 1 y 22 (22) = y (1) = y2 = y1 + h.f (21,91 1 -, 1.5 = 1.5 + hf 1 = 1.5 + - 2 0.255165123780745 = 1.62758256189037 Answer: y (1) = 1.62758256189037

1 Your Input: find y (1) for y' = -y + 2 cos (x), when y (0) = 1, h = using the Euler's method. The Euler's method states that Yn+1 = Yn +h. f (In, Yn), where Xn+1 = Xn + h. 1 We have that h Xo = 0,40 = 1, f(x, y) = -y + 2 cos (2) 4 Step 1. 1 1 X 1 = xo + h = 0 + 4 4 y (x1) = y = yı = yo +h. f (x0, yo) = 1+h: f(0, 1) = (6) 1 + 2 (1.0) = 1.25 1 = 1 + Step 2. 1 1 1 X2 = X1 +h + 4 4 2 1 y (22) = y 101 = y2 = yı +h. f (x1, yı) 1.25 + h:f +(6,1.25) – = 1.25 + (0.68782484342129) 1.42195621085532 4 Step 3. 1 1 3 X 3 = x2 + h = + - 2 4 4 3 y (x3) = y ܒܝ 1ܒܓܝ Y3 = y2 +hf (x2, Y2) = 1.42195621085532 +h 4 f 1.42195621085532 2 1 = 1.42195621085532 + 1.50525843908668 (0.333208912925423) 4 Step 4. 3 1 X 4 = X3 + h + = 1 4 4 y (x4) = y(1) = 44 = 4; +h: f (x3, 43) 1.50525843908668 +h 3 ·f 1.50525843908668 1 = 1.50525843908668 + (-0.0418807013390363) = 1.49478826375192 4 Answer: y (1) = 1.49478826375192
part(b)

1 Your input: find y (1) for y' = y - 2 sin (2), when y (0) = 1, h = using the Euler's method. 2 The Euler's method states that Yn+1 = Yn +h. f (xn, yn), where Xn+1 = xn +h. 1 We have that h = 20 = 0,90 = 1, f(x, y) = y - 2 sin (2) 2 Step 1. 1 1 X1 = xo +h=0+ = 1 y (x1) = y = y1 = yo +h: f (xo, yo) = 1+h: f(0, 1) = 2 = 1 + (1.0) = 1.5 2 Step 2. 1 1 X2 = X1 +h = + =1 y (x2) = y(1) = y2 = y; +h: f (x1, yı) = 1.5+hºf (5,1.5) – 1 = 1.5 + (0.541148922791594) 1.7705744613958 2 Answer: y (1) = 1.7705744613958

1 Your Input: find y (1) for y' = y — 2 sin (2), when y(0) = 1, h = using the Euler's method. 4 The Euler's method states that yn+1 = Yn + hºf (In, Yn), where Xn+1 = In +h. 1 We have that h X0 = 0,40 = 1, f(x, y) = y — 2 sin (2) 4 Step 1. 1 1 X1 = xo + h = 0+ 4 4 y (x1) y = y1 = yo +h. f (x0, yo) = 1+hºf (0,1) = =1+ (1.0) = = 1.25 4 Step 2. 1 1 1 X2 = X1 +h + = 4 4 2 1 y (x2) = y 6 = y) = x +h: f (311 3.) – 1.25 +2.5 (1.1.25) = 2 1 = 1.25 + . 4 (0.755192081490954) = 1.43879802037274 Step 3. 1 1 X3 = x2 +h + 2 Alco = 4 3 y (23) =y = y3 = y2 +h. f (x2, y2) = 1.43879802037274 +h . 21+ 4 -sch .,1.43879802037274 1 = 1.43879802037274 + (0.479946943164333) 1.55878475616382 4 Step 4. 1 X4 = X3 + h= 3 + 4. = 4 y (x4) = y(1) = 44 = y; +h: f (x3, 43) 1.55878475616382 +h ·f 1.55878475616382 1 = 1.55878475616382 + (0.195507236117153) 1.60766156519311 4 Answer: y (1) = 1.60766156519311
part(c)

1 Your Input: find y (1) for y' = -5y + 4 sin (x) + 6 cos (x), when y (0) = 1, h = using the Euler's method. 2 The Euler's method states that Yn+1 = Yn +hºf (&n, yn), where Xn+1 = xn +h. 1 We have that h = 20 = 2 0,40 = 1, f(x, y) = -5y + 4 sin (x) + 6 cos (x) Step 1. 1 1 X1 = xo + h = 0 + = 2 2 1 y (21) = y y () = y1 = yo +h: f(x0, yo) = 1+h: f(0, 1) = 2 1 =1+ (1.0) = 1.5 2 Step 2. 1 1 X2 = X1 +h + = 1 2 2 y (x2) = y(1) = y2 = y1 +h: f(x1, y1) 1.5 + h·f $(,15) - 1 = 1.5 + . (-0.316802474240952) = 1.34159876287952 2 Answer: y(1) = 1.34159876287952

1 Your Input: find y (1) for y' -5y + 4 sin(x) + 6 cos (x), when y (0) = 1, h = using the Euler's method. The Euler's method states that yn+1 = yn + h • f(xn, yn), where xn+1 = X n + h. We have that h = 20 4' 0,40 = 1, f(x, y) = -5y + 4 sin (x) + 6 cos (x) Step 1. 1 X1 = xo + h = 0 + 1 À 4 4 1 y (x1) = y = y1 = yo +hºf (x0, yo) = 1+h: f (0, 1) = 1 + · (1.0) = 1.25 4 Step 2. 1 1 1 X2 = X1 +h + 4 4 2 y (x2) = y FIS = y2 = yı +h·f (21, yı) = 1.25 +hºf *(5,1.25) 1 = 1.25 + 4 · (0.55309036728196) = 1.38827259182049 Step 3. 1 1 3 X3 X2 + h = + = 2 4 4 y (33) = y u = y3 = y2 +h: f (x2, Y2) = 1.38827259182049 +h 1.38827259182049 Aco - 2 = 1.38827259182049 + . (0.241834566656598) 1. 73123348464 4 Step 4. 3 1 X4 = x3 + h + = 1 4 4 y (24) = y(1) = 44 = y; +h. f (x3, Y3) = 1.44873123348464 +h 3 :f G 1.44873123348464 1 = 1.44873123348464 + . +1 (-0.126967914086937) = 1.41698925496291 Answer: y (1) = 1.41698925496291
part(d)

1 Your input: find y (1) for y' = 5y – 6 sin (x) – 4 cos (x), when y (0) = 1, h = using the Euler's method. 2 The Euler's method states that Yn+1 = Yn +h: f (xn, yn), where Xn+1 = In +h. We have that h = 20 = 0,40 = 1, f(x, y) = 5y – 6 sin (x) – 4 cos (x) 2 Step 1. 1 X1 = xo +h= 0 + = = 2 y (x1) = y = y1 = yo +h: f (xo, yo) = 1+h: f (0, 1) = 2 1 = 1 + + · (1.0) = 1.5 Step 2. 1 1 X2 = X1 +h = + = 1 2 2 y (x2) = y(1) = y2 = y1 +hºf (x1, y1) = 1.5+h:f -(1,1,5) = - 1.5 + 1. (1.11311652081329) = 2.05655826040665 Answer:y (1) = 2.05655826040665

1 Your input: find y (1) for y' = 5y - 6 sin (2) - 4 cos (2), when y (0) = 1, h = using the Euler's method. The Euler's method states that yn+1 = Yn+hºf (In, yn), where xn+1 = In +h. We have that h = À Xo = 0, yo = 1, f(x,y) = 5y – 6 sin (x) – 4 cos (x) Step 1. 1 1 x1 = x + h = 0 + - 4 4 y(x1) = =y A) = y1 = yo + hºf (20,y) = 1+h. f(0, 1) = = 1+ (1.0) = 1.25 4 Step 2. 1 1 X2 = x1 + h = = 4 ta 4 2 y (x2) = =y = y2 = y1 +hf (21, yı) = 1.25+hºf +(11:25) = N = 1.25 + . (0.889926557630283) = 1.47248163940757 4 Step 3. 1 3 X3 = 22 +h= + = 2 4 4 3 y(x3) = y = y3 = y2 +h. f (x2, Y2) = 1.47248163940757+h .f ( , 1.47248163940757 1 = 1.47248163940757 + (0.975524717851145) = 1.71636281887036 4 Step 4. 1 X 4 = x3 + h = 3 4 + = 1 4 y (x4) = y(1) = 44 = y; +h. f (23,43) = 1.71636281887036 +h .f (6,1.7 1.71636281887036 1 = 1.71636281887036 + . (1.5652260587165) = 2.10766933354948 4 Answer: y (1) = 2.10766933354948