32. Let yı and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(y1, y2). Show that dW A(x)- = (y1)(Ay") – (y2)(Ay{"). dx Then substitute for Ay and Ay" from the original differ- ential equation to show that dW A(x) dx = -B(x)W(x). (b) Solve this first-order equation to deduce Abel's for- mula W(x) = K exp (- | (-) B(x) dx A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W(y1, y2) is either zero every- where or nonzero everywhere (as stated in Theorem 3)?
32. Let yı and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(y1, y2). Show that dW A(x)- = (y1)(Ay") – (y2)(Ay{"). dx Then substitute for Ay and Ay" from the original differ- ential equation to show that dW A(x) dx = -B(x)W(x). (b) Solve this first-order equation to deduce Abel's for- mula W(x) = K exp (- | (-) B(x) dx A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W(y1, y2) is either zero every- where or nonzero everywhere (as stated in Theorem 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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need help please for 32

Transcribed Image Text:32. Let yı and y2 be two solutions of A(x)y" + B(x)y' +
C(x)y
are continuous and A(x) is never zero. (a) Let W =
W(y1, y2). Show that
= 0 on an open interval I where A, B, and C
dW
A(x)-
dx
= (v1)(Ay") – (y2)(Ay{).
Then substitute for Ay and Ay from the original differ-
ential equation to show that
dW
A(x)
3 — В (х) W(x).
dx
(b) Solve this first-order equation to deduce Abel's for-
mula
B(x)
W(x) = K exp (- /
AG dx).
where K is a constant. (c) Why does Abel's formula
imply that the Wronskian W(y1, y2) is either zero every-
where or nonzero everywhere (as stated in Theorem 3)?
Apply Theorems 5 and 6 to find general solutions of the dif-
ferential equations given in Problems 33 through 42. Primes
denote derivatives with respect to x.
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