" +p(t)y +q(t)y -0 (1), can be put in more suitable form for finding a solution by making a change of the independent and or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variabie Let z u (t) be the new independent variable. It is easy to show that - and - ()*+ , The differential equation becomes (告)+(+p(0%)+q(0y-0 (0). In order for equation (1) to have constant coefficients, the coefficients of and y must be proportional If q (t) >0, then choose the constant of proportionality to be 1. Hence, z = u (t) = Sla (t)i dt. With z chosen this way, the coefficient of in equation (i) is also a constant, provided that the function H = is a constant. Thus, equation (1) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function H is a constant. Try to transform the given equation into one with constant coefficients by this method. Is it possible? y" + 8ty +y = 0
" +p(t)y +q(t)y -0 (1), can be put in more suitable form for finding a solution by making a change of the independent and or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variabie Let z u (t) be the new independent variable. It is easy to show that - and - ()*+ , The differential equation becomes (告)+(+p(0%)+q(0y-0 (0). In order for equation (1) to have constant coefficients, the coefficients of and y must be proportional If q (t) >0, then choose the constant of proportionality to be 1. Hence, z = u (t) = Sla (t)i dt. With z chosen this way, the coefficient of in equation (i) is also a constant, provided that the function H = is a constant. Thus, equation (1) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function H is a constant. Try to transform the given equation into one with constant coefficients by this method. Is it possible? y" + 8ty +y = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![y" +p (t)y' +q(t)y = 0 (i),
can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show
(島)
( + ( +p(t) + a()y = 0 (1i).
dy
that
dt
dz dy
d'y
and
2 d'y
d²z dy
dt dz
The differential equation becomes
dt dr
dt
d'y
d'z
de
dz dy
d'y
, 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H =
g'(t) +2p(t)q(t)
is a
dy
In order for equation (ii) to have constant coefficients, the coefficients of
constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function H is a constant.
Try to transform the given equation into one with constant coefficients by this method. Is it possible?
y" + 8ty' + ty = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56c39839-d6ec-4cd3-94f9-6bb9a1cb44c1%2Fbea6c28f-ccc3-4be5-aceb-f99003f3d4fa%2F07g27pb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:y" +p (t)y' +q(t)y = 0 (i),
can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show
(島)
( + ( +p(t) + a()y = 0 (1i).
dy
that
dt
dz dy
d'y
and
2 d'y
d²z dy
dt dz
The differential equation becomes
dt dr
dt
d'y
d'z
de
dz dy
d'y
, 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H =
g'(t) +2p(t)q(t)
is a
dy
In order for equation (ii) to have constant coefficients, the coefficients of
constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function H is a constant.
Try to transform the given equation into one with constant coefficients by this method. Is it possible?
y" + 8ty' + ty = 0
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