" +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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

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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