Suppose that the three numbers r,, 12, and r3 are distinct. Show that the three functions exp (r,x), exp (12x), and exp (r3x) are linearly independent by showing that their Wronskian given below is nonzero for all x. 1 1 1 W= exp [(1 +12+r3)x]* 1 If y1. Y2. y, are n solutions of the homogeneous nth-order linear equation y) + p, (x)y" + Pn - 1(x)y'+P (x)y = 0 on an open interval I, where each p, is continuous. Then the Wronskian is W = W(y,, y2, .., yn). If y,, y2. ... yn are linearly independent, then W#0 at each point of I. To simplify W, evaluate the determinant. W= exp [(1 +12 +r3)*]•O
Suppose that the three numbers r,, 12, and r3 are distinct. Show that the three functions exp (r,x), exp (12x), and exp (r3x) are linearly independent by showing that their Wronskian given below is nonzero for all x. 1 1 1 W= exp [(1 +12+r3)x]* 1 If y1. Y2. y, are n solutions of the homogeneous nth-order linear equation y) + p, (x)y" + Pn - 1(x)y'+P (x)y = 0 on an open interval I, where each p, is continuous. Then the Wronskian is W = W(y,, y2, .., yn). If y,, y2. ... yn are linearly independent, then W#0 at each point of I. To simplify W, evaluate the determinant. W= exp [(1 +12 +r3)*]•O
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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![Suppose that the three numbers r,, r2, and r3 are distinct. Show that the three functions exp (r,x), exp (r2x), and exp (13x)
are linearly independent by
showing that their Wronskian given below is nonzero for all x.
1
1
1
W= exp [(1 +12 +r3) x]•
1 2 '3
1
2
3
is
If y1. Y2. . yn are n solutions of the homogeneous nth-order linear equation y" +p, (x)y" "+... + pn- 1(x)y' + P, (x)y = 0 on an open interval I, where each
Pi
continuous. Then the Wronskian is W= W(y,. y2:
Yn). If y1.
1: Y2:
Y, are linearly independent, then W#0 at each point of I.
---:
To simplify W, evaluate the determinant.
W= exp [(1 +2 +r3) x] •
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Transcribed Image Text:Suppose that the three numbers r,, r2, and r3 are distinct. Show that the three functions exp (r,x), exp (r2x), and exp (13x)
are linearly independent by
showing that their Wronskian given below is nonzero for all x.
1
1
1
W= exp [(1 +12 +r3) x]•
1 2 '3
1
2
3
is
If y1. Y2. . yn are n solutions of the homogeneous nth-order linear equation y" +p, (x)y" "+... + pn- 1(x)y' + P, (x)y = 0 on an open interval I, where each
Pi
continuous. Then the Wronskian is W= W(y,. y2:
Yn). If y1.
1: Y2:
Y, are linearly independent, then W#0 at each point of I.
---:
To simplify W, evaluate the determinant.
W= exp [(1 +2 +r3) x] •
%3D
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