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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
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 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 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
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.
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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