where A system of system of first-order linear differential equations has the form y' = Ay 02n ⠀ ann, Solve the system of linear differential equations by following the steps listed. 9/1 3/2 012 022 ⠀ and an2 a₁1 021 This gives us the system 3/₁ = y1 - 4y2 3/2 = -2y1 + 8y2 ... (a) We first want to write this as a matrix equation y'= Ay. List out what the matrix A should be in this matrix equation. ain (b) We would like to have the equations so that y is a function of only y₁ and 2 is a function of only 32. To do this, we want to turn A into a diagonal matrix. This can be done by taking the diagonalization of A. Diagonalize the matrix A you created in part a. (c) Now that we have a matrix P such that P-¹AP = D is diagonal, we substitute y' and y to be Pw' = y Pw = y Pw' = APW Since P is invertible, we can multiply P-1 to the left on both sides of the equations to get w' = P-¹APW = Dw Write out what the system of linear differential equations looks like now.
where A system of system of first-order linear differential equations has the form y' = Ay 02n ⠀ ann, Solve the system of linear differential equations by following the steps listed. 9/1 3/2 012 022 ⠀ and an2 a₁1 021 This gives us the system 3/₁ = y1 - 4y2 3/2 = -2y1 + 8y2 ... (a) We first want to write this as a matrix equation y'= Ay. List out what the matrix A should be in this matrix equation. ain (b) We would like to have the equations so that y is a function of only y₁ and 2 is a function of only 32. To do this, we want to turn A into a diagonal matrix. This can be done by taking the diagonalization of A. Diagonalize the matrix A you created in part a. (c) Now that we have a matrix P such that P-¹AP = D is diagonal, we substitute y' and y to be Pw' = y Pw = y Pw' = APW Since P is invertible, we can multiply P-1 to the left on both sides of the equations to get w' = P-¹APW = Dw Write out what the system of linear differential equations looks like now.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 31E
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![where
A system of system of first-order linear differential equations has the form
y' = Ay
y' =
, y =
Y1
Y2
, A =
a11
a21
This gives us the system
B
[an an2
ann]
Solve the system of linear differential equations by following the steps listed.
y₁ = y₁ 432
y₂ = -2y1 + 8y2
a12
022
ain
a2n
⠀
(a) We first want to write this as a matrix equation y'= Ay. List out what the matrix A
should be in this matrix equation.
(b) We would like to have the equations so that y is a function of only y₁ and ₂2 is a function
of only y2. To do this, we want to turn A into a diagonal matrix. This can be done by
taking the diagonalization of A. Diagonalize the matrix A you created in part a.
(c) Now that we have a matrix P such that P-¹AP = D is diagonal, we substitute y' and
y to be
Pw' = y
Pw=y
Pw = APw
Since P is invertible, we can multiply P-1 to the left on both sides of the equations to
get
w = P-¹APW = Dw
Write out what the system of linear differential equations looks like now.
(d) Let k be any scalar. A general solution to the equation y'= ky(t) is
y(t) = Cekt
where C is a constant from integration. Use this to solve the two equations you have
made in part c. Note that it will not be the same C for each equation, you need to
differentiate the C's in each solution which you can do by using C₁ and C₂.
(e) Our original problem was in terms of y, not w. So we can finish solving this system by
solving y = Pw. You found P in part b and w in part d. Multiply this matrix and vector
together to get a final solution to the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39ee3059-6733-456c-bbc4-46bedd3c18f2%2Fea94e7dd-1a82-4c22-a12a-f02691d7bca8%2Fndbwpyl_processed.png&w=3840&q=75)
Transcribed Image Text:where
A system of system of first-order linear differential equations has the form
y' = Ay
y' =
, y =
Y1
Y2
, A =
a11
a21
This gives us the system
B
[an an2
ann]
Solve the system of linear differential equations by following the steps listed.
y₁ = y₁ 432
y₂ = -2y1 + 8y2
a12
022
ain
a2n
⠀
(a) We first want to write this as a matrix equation y'= Ay. List out what the matrix A
should be in this matrix equation.
(b) We would like to have the equations so that y is a function of only y₁ and ₂2 is a function
of only y2. To do this, we want to turn A into a diagonal matrix. This can be done by
taking the diagonalization of A. Diagonalize the matrix A you created in part a.
(c) Now that we have a matrix P such that P-¹AP = D is diagonal, we substitute y' and
y to be
Pw' = y
Pw=y
Pw = APw
Since P is invertible, we can multiply P-1 to the left on both sides of the equations to
get
w = P-¹APW = Dw
Write out what the system of linear differential equations looks like now.
(d) Let k be any scalar. A general solution to the equation y'= ky(t) is
y(t) = Cekt
where C is a constant from integration. Use this to solve the two equations you have
made in part c. Note that it will not be the same C for each equation, you need to
differentiate the C's in each solution which you can do by using C₁ and C₂.
(e) Our original problem was in terms of y, not w. So we can finish solving this system by
solving y = Pw. You found P in part b and w in part d. Multiply this matrix and vector
together to get a final solution to the system.
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