I asked ChatGPT to solve the second order linear non-homogeneous equation y"+2y+y= Is the solution that ChatGPT offers correct, or is there one or more mistakes in the work? If the work is correct, verify the solution. If it is incorrect, identify the exact place(s) where the mistake(s) occur. Explain your answer clearly! = 2x Below is the response from ChatGPT: Yes, I can solve the equation y"+2y+y = 2x. One way to solve it is to use the characteristic equation method. To find the characteristic equation, we set the left side of the equation to zero and solve for the roots of the characteristic equation, which is given by: r² + 2r + 1 = 0 Solving for the roots, we get: r₁=1+√-1-4--1+√-5 and r₂ =-1-√√-1-4 =-1-√√-5 So the roots are complex conjugates, which means the general solution of the homogeneous equation will be of the form: Yn(x) = e¯ª (c₁ cos (√5x) + c2 sin (√5 . x)) Where c₁ and c₂ are arbitrary constants. To find a particular solution, we can use the method of undetermined coefficients. Since the non- homogeneous term 2x is a polynomial of degree 1, we can assume that the particular solution is of the form: yp(x)= = ax + b Substituting this into the original equation and equating the coefficients of the like powers of a gives: a = 2 and b = 0 Therefore the particular solution is
I asked ChatGPT to solve the second order linear non-homogeneous equation y"+2y+y= Is the solution that ChatGPT offers correct, or is there one or more mistakes in the work? If the work is correct, verify the solution. If it is incorrect, identify the exact place(s) where the mistake(s) occur. Explain your answer clearly! = 2x Below is the response from ChatGPT: Yes, I can solve the equation y"+2y+y = 2x. One way to solve it is to use the characteristic equation method. To find the characteristic equation, we set the left side of the equation to zero and solve for the roots of the characteristic equation, which is given by: r² + 2r + 1 = 0 Solving for the roots, we get: r₁=1+√-1-4--1+√-5 and r₂ =-1-√√-1-4 =-1-√√-5 So the roots are complex conjugates, which means the general solution of the homogeneous equation will be of the form: Yn(x) = e¯ª (c₁ cos (√5x) + c2 sin (√5 . x)) Where c₁ and c₂ are arbitrary constants. To find a particular solution, we can use the method of undetermined coefficients. Since the non- homogeneous term 2x is a polynomial of degree 1, we can assume that the particular solution is of the form: yp(x)= = ax + b Substituting this into the original equation and equating the coefficients of the like powers of a gives: a = 2 and b = 0 Therefore the particular solution is
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:I asked ChatGPT to solve the second order linear non-homogeneous equation
y"+2y+y= = 2x
Is the solution that ChatGPT offers correct, or is there one or more mistakes in the work? If the work is
correct, verify the solution. If it is incorrect, identify the exact place(s) where the mistake(s) occur.
Explain your answer clearly!
Below is the response from ChatGPT:
Yes, I can solve the equation y"+2y+y = 2x. One way to solve it is to use the characteristic equation
method. To find the characteristic equation, we set the left side of the equation to zero and solve for the
roots of the characteristic equation, which is given by:
r² + 2r + 1 = 0
Solving for the roots, we get:
r₁=1+√-1-4--1+√-5
and
r₂ =-1-√√-1-4 =-1-√√-5
So the roots are complex conjugates, which means the general solution of the homogeneous equation will
be of the form:
Yn(x) = e¯² (c₁ cos ( √5x) + ₂ sin (√5. x))
Where c₁ and c₂ are arbitrary constants.
To find a particular solution, we can use the method of undetermined coefficients. Since the non-
homogeneous term 2x is a polynomial of degree 1, we can assume that the particular solution is of the
form:
yp(x) = ax + b
Substituting this into the original equation and equating the coefficients of the like powers of a gives:
a = 2 and b = 0
Therefore the particular solution is
Yp(x) = 2x
The general solution of the non-homogeneous equation is then given by:
(c₁ cos (√52)+c₂ sin (√5.x)) +
C1
y(x)
=
=
-x
Yh(x) + Yp(x) *
= e
where c₁ and c₂ are arbitrary constants of integration.
+ 2x
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