Let u₁ and u2 be two linearly independent solutions of the second-order linear differential equation (i) (ii) d²u dx² 2 fu + = Let w = u2/u1. Show that w' = c/u for some nonzero constant c. -2u1/u₁. Show that y satisfies the Ricatti equation Let y = y' - 0, ƒ = f (x). y² 2 - f.

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 54E: Plant Growth Researchers have found that the probability P that a plant will grow to radius R can be...
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Let u₁ and u2 be two linearly independent solutions of the second-order
linear differential equation
(i)
(ii)
d² u
fu
dx² 2
+
-
0, f = f(x).
Let w = uz/u₁. Show that w' = c/u for some nonzero constant c.
Let y = -2u₁/u₁. Show that y satisfies the Ricatti equation
y²
V-2-1
y'
f.
Transcribed Image Text:Let u₁ and u2 be two linearly independent solutions of the second-order linear differential equation (i) (ii) d² u fu dx² 2 + - 0, f = f(x). Let w = uz/u₁. Show that w' = c/u for some nonzero constant c. Let y = -2u₁/u₁. Show that y satisfies the Ricatti equation y² V-2-1 y' f.
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ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,