As a spring is heated, its spring "constant" decreases. Suppose the spring is heated so that the spring "constant" at time t is k(t)=7-t N/m. If the unforced mass-spring system has mass m = 2 kg and a damping constant b = 1 N-sec/m with initial conditions x(0) = 4 m and x'(0) = 0 m/sec, then the displacement x(t) is governed by the initial value problem 2x'' (t) + x'(t)+(7-t)x(t) = 0; x(0) = 4, x'(0) = 0. Find the first four nonzero terms in a power series expansion about t=0 for the displacement.
As a spring is heated, its spring "constant" decreases. Suppose the spring is heated so that the spring "constant" at time t is k(t)=7-t N/m. If the unforced mass-spring system has mass m = 2 kg and a damping constant b = 1 N-sec/m with initial conditions x(0) = 4 m and x'(0) = 0 m/sec, then the displacement x(t) is governed by the initial value problem 2x'' (t) + x'(t)+(7-t)x(t) = 0; x(0) = 4, x'(0) = 0. Find the first four nonzero terms in a power series expansion about t=0 for the displacement.
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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The last two questions in the second image is important which should be answered, the first image is about power series expansion scenario.
Transcribed Image Text:As a spring is heated, its spring "constant" decreases. Suppose the spring is heated so that the spring "constant" at time t is k(t) = 7 - t N/m. If the unforced mass-spring system has mass
m = 2 kg and a damping constant b = 1 N-sec/m with initial conditions x(0) = 4 m and x'(0) = 0 m/sec, then the displacement x(t) is governed by the initial value problem
2x'' (t) + x' (t) + (7 − t)x(t) = 0; x(0) = 4, x'(0) = 0. Find the first four nonzero terms in a power series expansion about t=0 for the displacement.
x(t) = + ...
(Type an expression that includes all terms up to order 4.)
k(t)= 7-t
m
heat
2 kg
1 N-sec/m
x(t)
x(0) = 4
x'(0) = 0
Transcribed Image Text:Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation.
(x² +19)y"+y=0
y(x) =
(Type an expression in terms of a and a₁ that includes all terms up to order 3.)
+ ...
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