Often a differential equation with variable coefficients, y" + p(t)y' + q(t)y = 0 (i), can be transformed into an equation with constant coefficients by a change of the independent variable. Let x = u(t) = f(g(t)) ¹/2 dit, with q(t) > 0, be the new independent variable. If the function H q'(t) + 2p(t)q(t) 2(g(t))3/2 is a constant, then (i) can be transformed into an equation with constant coefficients by a change of the independent variable. Consider the differential equation y" + 4ty' + t²y = 0. Calculate H using the formula above, and then determine whether it is possible to transform the differential equation into one with constant
Often a differential equation with variable coefficients, y" + p(t)y' + q(t)y = 0 (i), can be transformed into an equation with constant coefficients by a change of the independent variable. Let x = u(t) = f(g(t)) ¹/2 dit, with q(t) > 0, be the new independent variable. If the function H q'(t) + 2p(t)q(t) 2(g(t))3/2 is a constant, then (i) can be transformed into an equation with constant coefficients by a change of the independent variable. Consider the differential equation y" + 4ty' + t²y = 0. Calculate H using the formula above, and then determine whether it is possible to transform the differential equation into one with constant
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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Transcribed Image Text:Often a differential equation with variable coefficients,
y" + p(t)y' + q(t)y = 0 (i),
can be transformed into an equation with constant coefficients by a
change of the independent variable. Let
x = u(t) = f(g(t)) ¹/² dt,
with q(t) > 0, be the new independent variable. If the function
H
q'(t) + 2p(t)q(t)
2(g(t))3/2
is a constant, then (i) can be transformed into an equation with
constant coefficients by a change of the independent variable.
Choose one
Consider the differential equation y" + 4ty' + t²y = 0. Calculate H
using the formula above, and then determine whether it is possible
to transform the differential equation into one with constant
coefficients using this method.
H =
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