Query_11

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course MTH-279

06/25 around 11:50AM.

Question: 3.8.4. Solve the equation y ' = - y + t with y(0) = 0.Write the expression y_(k + 1) = y_k + h f (t_k, y_k) for h = .01.

Find y_k for k = 0, 1, 2, 3.

Using your original solution for the equation, compare your values of y_k with the values given by the accurate solution.

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Your solution:

y' + y = t

u(t) = e^(int p(t) dt) = e^t

y'e^t + ye^t = te^t

(ye^t)' = te^t

Integration yeilds: (udv = uv - int(vdu) for the second half) u = t, du = dt, dv = e^t, v = e^t

ye^t = (te^t - e^t) + c

y(t) = (t-1) + c/e^t

y(0) = 0

y(0) = (0-1) + c/e^0

= -1 + c

c = 1

y(t) = (t-1)+1/e^t

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Actual solution using: y(t) = (t-1)+1/e^t

t | y

0 0 | 0

1 0.01 | 4.98E-5

2 0.02 | 1.99E-4

3 0.03 | 4.46E-4

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Using Euler's Method: y' = f(t,y) = -y + t

t | y

0 0 | 0

1 0.01 | 0

2 0.02 | 1E-4

3 0.03 | 2.99E-4

y1 = y0 + 0.01*f(t0,y0) = 0 + 0.01(0+0) = 0

y2 = y1 + 0.01*f(t1,y1) = 0 + 0.01(-0+0.01) = 1E-4

y3 = y2 + 0.01*f(t2,y2) = 1E-4 + 0.01(-1E-4 + 0.02) = 2.99E=4

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

Self-critique rating:

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Question: 3.8.6. Euler's method applied to the equation y ' = alpha t + beta, y(t_0) = y_0, yields y values -1, -.9, -.81 and -.73 at respective t values 0, .1, .2, .3. Find the values of alpha, beta, t_0 and y_0.

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Your solution:

y' = At + B y(t0) = y0

t | y

0 0 | -1

1 0.1 | -0.9

2 0.2 | -0.81

3 0.3 | -0.73

h = 0.1

t_0 = 0

y_0 = -1

y_k+1 = y_k + 0.1*f(t,y)

f(t,y) = At+B

y1 = y0 + 0.1*(At+B)

-0.9 = -1 + 0.1*(At+B)

-0.9 = -1 + 0.1*(B) -> t_0 = 0

0.1 = 0.1 B

B = 1

y2 = y1 + 0.1(At+B)

-0.81 = -0.9+0.1(At+B)

-0.81 = -0.9+0.1(A(0.1)+1)

0.09 = 0.1(A(0.1)+1)

0.9 = A * 0.1 + 1

-0.1 = A * 0.1

A = -1

Confidence rating: 2

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Given Solution:

Self-critique (if necessary):

Self-critique rating:

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Question: 3.8.8a. For each of the following situations, will Euler's method overestimate or underestimate the values of the solution to an equation:

• The solution curve is known to be increasing and concave up.

• The solution curve is known to be increasing and concave down.

• The solution curve is known to be decreasing and concave up.

• The solution curve is known to be decreasing and concave down.

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Your solution:

Your solution:

A - Underestimate

B - Overestimate

C - Underestimate

D - Overestimate

why?

Concave up = under

concave down or convex = over

Euler's method is a straight-line prediction, using tangent lines

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Given Solution:

Self-critique (if necessary):

Self-critique rating:

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Question: 3.8.14. y ' = y^2 with y(0) = 1.

Solve the equation.

Perform Euler's Method to approximate the values of the solution on the t interval [0, 1.2] with step size h = .1.

Compare the values you get with the values given by your solution to the equation.

This could be done by hand, but it would take awhile and the probability of an error would be relatively high. A spreadsheet is recommended.

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Your solution:

y' = y^2 y(0) = 1

dy/y^2 = 1 dt

y^-2 dy = 1 dt

Integration yeilds:

-y^-1 = t + c

-1/y = t + c

y = -1/(t + c)

y(0) = 1

y(0) = -1/(0 + c) = 1

-1/c = 1

c = -1

y(t) = -1/(t-1)

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Actual using: y(t) = -1/(t-1)

t | y

0 0 | 1

1 0.1 | 1.11

2 0.2 | 1.25

3 0.3 | 1.429

4 0.4 | 1.67

5 0.5 | 2

6 0.6 | 2.5

7 0.7 | 3.33

8 0.8 | 5

9 0.9 | 10

10 1.0 |error (Asymptote) 1-1 = 0

11 1.1 | -10

12 1.2 | -5

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Euler's Method: y' = y^2

t | y

0 0 | 1

1 0.1 | 1.11

2 0.2 | 1.221

3 0.3 | 1.37

4 0.4 | 1.56

5 0.5 | 1.8

6 0.6 | 2.125

7 0.7 | 2.58

8 0.8 | 3.24

9 0.9 | 4.29

10 1.0 | 6.12

11 1.1 | 9.885

12 1.2 | 19.66

y_1 = y_0 + 0.01y_0^2

y_2 = y_1 + 0.01y_1^2

confidence rating #$&*:

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&#Your work looks very good. Let me know if you have any questions. &#