QUESTION 1 (n) be solutions of the homogeneous nth-order differential equation a (x) y n Let y ₁₂ k 1 Then the linear combination y=c v + c 1 1 22 interval. this is according to the principle of linear independence Wronskian superposition existence and uniqueness + ...+ where the c k + a n-1 (x) y (n-1) + .+a₁(x)y'+a (x)y=0 , i=1, 2, 3, ..., k are arbitrary constants, is also a solution on the ¡³
QUESTION 1 (n) be solutions of the homogeneous nth-order differential equation a (x) y n Let y ₁₂ k 1 Then the linear combination y=c v + c 1 1 22 interval. this is according to the principle of linear independence Wronskian superposition existence and uniqueness + ...+ where the c k + a n-1 (x) y (n-1) + .+a₁(x)y'+a (x)y=0 , i=1, 2, 3, ..., k are arbitrary constants, is also a solution on the ¡³
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please answer the question below thank you.
Transcribed Image Text:QUESTION 1
Let y
(n)
(n-1)
+ a
12. be solutions of the homogeneous nth-order differential equation a (x) y
y
k
(x) y +...+ a
+ a₁(x)y'+a₂(x)y=0
n
n-1
+c.y. where the c i=1, 2, 3, ..., k are arbitrary constants, is also a solution on the
k
Then the linear combination y=c_y +c√√₂ +
interval. this is according to the principle of
O linear independence
O Wronskian
O superposition
O existence and uniqueness
QUESTION 2
3
The functions y₁= x³ and y₁=x4 are solutions of .x²y " − 6xy' + 12y = 0 on the interval (0, ∞ ). Which of the following is not true?
1
|y₁=
=cy
O The Wonskian of y, and y are not equal to zero.
1
2
The linear combination of y, and y, is also a solution that is y=c₁v₁ + cy₁₂²
1
2
y, and y are linearly independent
2
Transcribed Image Text:QUESTION 3
A set of n functions f. (x),
²₁(x), ƒ₂(x), ..., ƒ„‚(x) is
of the remaining functions.
O linearly dependent
O linearly independent
QUESTION 4
on an interval / if at least one of the functions can be expressed as linear combination
Given ƒ₁(x), ƒ₂(x), ƒ 3(x), ƒ4(x) which make up a set. Which of the following describes the set being linearly dependent?
Of ₂(x) = c₂f₁(x) + c
² 2£3 (x) + C 3f 4 ( x)
c
1
○ f(x) = f₁(x) + 2ƒ₂(x) − 3ƒ z(x) +ƒ 4(x)
O the Wonskian is not equal to zero
O The functions are not multiples of each other.
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