Consider any(n) + an-1y'An:n-1y(n-1)+ a1y' + aoy = f(x) where an, ...a1, ao E R, where an # 0....d. Rewrite the left-hand side of the equation as A(y), where A is a linear(a) Let D =operator. (Hint: Write A in terms of D)dx(b) In order to solve this ODE using Method of Undetermined Coefficients, we must be ableto find another linear operator L such that L(f) =of D. For what types of functions f can we find such an operator L? Explain.= 0. Note that L must also be in terms(c) Suppose we can find L such that L(f)= 0. What is Lo A?(d) Construct an example of a constant-coefficient, linear, non-homogeneous ODE. Find Land solve by applying L to both sides of the equation. Note that this method is essentiallythe rigorous version of Method of Undetermined Coefficients.

consider anyn+an-1yn-1+.....+a1y'+a0y=fx ...i where an, ......., a1 , a0∈ℝ where an≠0 (a)…

Answered: Consider any(n) + an-1y(n-1) + + a1y' +…

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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Related questions

Question

Consider any(n) + an-1y'
An:
n-1y(n-1)
+ a1y' + aoy = f(x) where an, ...a1, ao E R, where an # 0.
...
d
. Rewrite the left-hand side of the equation as A(y), where A is a linear
(a) Let D =
operator. (Hint: Write A in terms of D)
dx
(b) In order to solve this ODE using Method of Undetermined Coefficients, we must be able
to find another linear operator L such that L(f) =
of D. For what types of functions f can we find such an operator L? Explain.
= 0. Note that L must also be in terms
(c) Suppose we can find L such that L(f)= 0. What is Lo A?
(d) Construct an example of a constant-coefficient, linear, non-homogeneous ODE. Find L
and solve by applying L to both sides of the equation. Note that this method is essentially
the rigorous version of Method of Undetermined Coefficients.

Expert Solution

Step 1

consider $a_{n} y^{(n)} + a_{n - 1} y^{(n - 1)} + . . . . . + a_{1} y' + a_{0} y = f (x) . . . (i)$

where $a_{n}, . . . . . . ., a_{1}, a_{0} \in ℝ w h e r e a_{n} \neq 0$

(a) let $D = \frac{d}{d x}$

given equation can be written as

$a_{n} \frac{d^{n} y}{d x^{n}} + a_{n - 1} \frac{d^{n - 1} y}{d x^{n - 1}} + . . . . . + a_{1} \frac{d y}{d x} + a_{0} y = f (x) \Rightarrow (a_{n} \frac{d^{n}}{d x^{n}} + a_{n - 1} \frac{d^{n - 1}}{d x^{n - 1}} + . . . . . + a_{1} \frac{d}{d x} + a_{0}) y = f (x)$