Given the differential equation y' = 8y2 and y (0) = - 1 and its next few implicit derivatives y" = 16y y' y"' = 16 (y')2 + 16y y" y4 = 48 y'y" + 16 y y"" %3D Use the initial value "y(0) = -1" to find the value of "y'(0)", "y"(0)", "y"(0)", and "y (0)", then plug them into the Taylor Series Polynomial formula. 32 16 x2 + 2! 64 A y(x) 8 x! + 1! x° + x3 + x4 ... O! 3! 4! 8. 128 x2 128 384 By(x). O! x° + x! 1! 2! 3! 4! - 16 x2 + 112 x + 3! -4352 y(x). x! + 1! O! 2! 4! - 98304 8. x' + 1! -128 3072 Dy(x)= 0! x2 + x* +... 2! 3! 4! 2176 83968 Ey(x) 8. x' + 1! 128 x2 + x° + 0! 2! 3! 4! 480 960 F y(x) x° + O! x' + 1! x + 3! | ... 2! 4! + + + +
Given the differential equation y' = 8y2 and y (0) = - 1 and its next few implicit derivatives y" = 16y y' y"' = 16 (y')2 + 16y y" y4 = 48 y'y" + 16 y y"" %3D Use the initial value "y(0) = -1" to find the value of "y'(0)", "y"(0)", "y"(0)", and "y (0)", then plug them into the Taylor Series Polynomial formula. 32 16 x2 + 2! 64 A y(x) 8 x! + 1! x° + x3 + x4 ... O! 3! 4! 8. 128 x2 128 384 By(x). O! x° + x! 1! 2! 3! 4! - 16 x2 + 112 x + 3! -4352 y(x). x! + 1! O! 2! 4! - 98304 8. x' + 1! -128 3072 Dy(x)= 0! x2 + x* +... 2! 3! 4! 2176 83968 Ey(x) 8. x' + 1! 128 x2 + x° + 0! 2! 3! 4! 480 960 F y(x) x° + O! x' + 1! x + 3! | ... 2! 4! + + + +
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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9
Transcribed Image Text:Given the differential equation
y' = 8y2 and y(0) = - 1
and its next few implicit derivatives
y" = 16y y'
%3D
y"' = 16 (y')2 + 16y y"
y = 48 y'y" + 16 y y""
Use the initial value "y(0) = -1" to find the value of "y'(0)", "y"(0)", "y"(0)", and "y (0)", then plug them into the Taylor Series
Polynomial formula.
A y(x) = 0 +
O!
16
64
8.
x' +
1!
32
+
3!
+
-
|
2!
4!
8.
x! +
1!
128
x2 +
128
x3 +
3!
384
B y(x) = 0
...
O!
2!
4!
- 16
- 4352
© y(x)
O!
8.
x' +
112
r3
+
...
1!
2!
3!
4!
- 128
x2 +
3072
x3 +
3!
- 98304
Dy(x)
= - 0 +
x! +
x+ +...
O!
1!
2!
4!
128
x2 +
2!
2176
83968
® y(x) =
O!
8
x!
+3
3!
(E
r+ + ...
1!
4!
O y(x) = 0 +
O!
960
480
x2
x! +
1!
x +
3!
+
+
...
2!
4!
+
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