Below are several homogeneous second-order linear differential equations (HSOLDES). You are given one solution to the homogeneous equation and will be asked to find a second linearly independent solution. You will also be asked to identify P(x) and calculate the Wronskian Functional [P]. Check the given solution is actually a solution to the given equation before proceeding. cos cos Remember SOLDE Video 01. The example found y(x) = 0, but we took y2(x)=COS without the negatives we drop multiplicative constants. √x a. y"+ 2y = 0.given y₁= cos(kx). (Answer entry help: notice k and x have a multiplication symbol between them. Remember you can enter that product as kex, or as k x, with a space between k and x.) P(x)=0 +[P]- k 72-sin(kx) b. y + 4y+3y=0.given y P(x)= 4 +[P]-e-4x 72 e -3r c. (1-x²) - 2xy + 2y=0.given y₁=x. P(x)= +[P]= 1-x² 1-x² 72-x-m(x+1)-x-m(x-1)-2
Below are several homogeneous second-order linear differential equations (HSOLDES). You are given one solution to the homogeneous equation and will be asked to find a second linearly independent solution. You will also be asked to identify P(x) and calculate the Wronskian Functional [P]. Check the given solution is actually a solution to the given equation before proceeding. cos cos Remember SOLDE Video 01. The example found y(x) = 0, but we took y2(x)=COS without the negatives we drop multiplicative constants. √x a. y"+ 2y = 0.given y₁= cos(kx). (Answer entry help: notice k and x have a multiplication symbol between them. Remember you can enter that product as kex, or as k x, with a space between k and x.) P(x)=0 +[P]- k 72-sin(kx) b. y + 4y+3y=0.given y P(x)= 4 +[P]-e-4x 72 e -3r c. (1-x²) - 2xy + 2y=0.given y₁=x. P(x)= +[P]= 1-x² 1-x² 72-x-m(x+1)-x-m(x-1)-2
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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![Below are several homogeneous second-order linear differential equations (HSOLDES). You are given one solution to the homogeneous equation and will be asked to find a second linearly independent solution. You will also be asked to identify P(x) and calculate the Wronskian Functional [P]. Check the given solution is actually a solution
to the given equation before proceeding.
cos x
Remember SOLDE Video 01. The example found y2(x) = -x , but we took y₂(x)=√x without the negative; we drop multiplicative constants.
a. y" + k²y = 0, given y₁ = cos(kx). (Answer entry help: notice k and x have a multiplication symbol between them. Remember you can enter that product as k*x, or as k x, with a space between k and x.)
P(x) = 0
[P] = k
Y2 =
b. y" + 4y' + 3y = 0, given y₁ = e.
P(x) = 4
+ [P] =
Y2 =
sin(kx)
P(x)=
+ [P]=
₂-4x
c. (1-x2) y" - 2x y' + 2 y = 0, given y₁ = x.
e-3x
2x
1-x
C
1-x
2
v2 = x-m(x+1)-x-m(x-1)-2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8e0ad34-47bd-4edd-8e64-7fbb4be3cef6%2F127f682c-73ac-4bf2-b470-3ba512176118%2Fr5f7bl_processed.png&w=3840&q=75)
Transcribed Image Text:Below are several homogeneous second-order linear differential equations (HSOLDES). You are given one solution to the homogeneous equation and will be asked to find a second linearly independent solution. You will also be asked to identify P(x) and calculate the Wronskian Functional [P]. Check the given solution is actually a solution
to the given equation before proceeding.
cos x
Remember SOLDE Video 01. The example found y2(x) = -x , but we took y₂(x)=√x without the negative; we drop multiplicative constants.
a. y" + k²y = 0, given y₁ = cos(kx). (Answer entry help: notice k and x have a multiplication symbol between them. Remember you can enter that product as k*x, or as k x, with a space between k and x.)
P(x) = 0
[P] = k
Y2 =
b. y" + 4y' + 3y = 0, given y₁ = e.
P(x) = 4
+ [P] =
Y2 =
sin(kx)
P(x)=
+ [P]=
₂-4x
c. (1-x2) y" - 2x y' + 2 y = 0, given y₁ = x.
e-3x
2x
1-x
C
1-x
2
v2 = x-m(x+1)-x-m(x-1)-2
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