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1. Let y, and y2 be solutions of the differential equation y" + P(x)y' + Q(x)y = 0. (1) %3D (We will assume that P and Q are continuous on any interval.) Since y1 and y2 are solutions of (1), we have yi + P(x)y1 + Q(x)yı = 0, (2) 2 + P(x)y½ + Q(x)y2 = 0. (3) (a) From equations (2) and (3), eliminate the terms Q(x)yı and Q(x)y2 to form a single equation. (b) Let W(x) = W (y1 , Y2)(x) be the Wronskian of y1 and y2 · Show that W' = %3D (c) Use the results in parts (a) and (b) to form a first-order differential equation in W.

1. Let y, and y2 be solutions of the differential equation y" + P(x)y' + Q(x)y = 0. (1) %3D (We will assume that P and Q are continuous on any interval.) Since y1 and y2 are solutions of (1), we have yi + P(x)y1 + Q(x)yı = 0, (2) 2 + P(x)y½ + Q(x)y2 = 0. (3) (a) From equations (2) and (3), eliminate the terms Q(x)yı and Q(x)y2 to form a single equation. (b) Let W(x) = W (y1 , Y2)(x) be the Wronskian of y1 and y2 · Show that W' = %3D (c) Use the results in parts (a) and (b) to form a first-order differential equation in W.

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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Let y1 and y2 be solutions of the differential equation y" + P(x)y' + Q(r)y = 0. (1) (We will assume that P and Q are continuous on any interval.) Since y1 and y2 are solutions of (1), we have yK + P(x)y½ + Q(x)yı = 0, (2) 2 + P(x)y½ + Q(x)y2 2 = 0. (3) (a) From equations (2) and (3), eliminate the terms Q(r)yı and Q(r)y2 to form a single equation. (b) Let W(x) = W(y1, 42)(x) be the Wronskian of y1 and y2. Show that W' = y1 – {y2 . (c) Use the results in parts (a) and (b) to form a first-order differential equation in W. (d) Solve the first-order DE from part (c) to obtain a formula for W.
Transcribed Image Text:Let y1 and y2 be solutions of the differential equation y" + P(x)y' + Q(r)y = 0. (1) (We will assume that P and Q are continuous on any interval.) Since y1 and y2 are solutions of (1), we have yK + P(x)y½ + Q(x)yı = 0, (2) 2 + P(x)y½ + Q(x)y2 2 = 0. (3) (a) From equations (2) and (3), eliminate the terms Q(r)yı and Q(r)y2 to form a single equation. (b) Let W(x) = W(y1, 42)(x) be the Wronskian of y1 and y2. Show that W' = y1 – {y2 . (c) Use the results in parts (a) and (b) to form a first-order differential equation in W. (d) Solve the first-order DE from part (c) to obtain a formula for W.
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