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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)?

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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.
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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