dra -AR=0, where A is a separation constant and R(r) is the radial factor in the solution. + In exercise LP 3, you found that R) and R), with (LP eqn 8) √1 + 4X 1 1/₂ = -3± were both solutions of the radial ODE. In this problem, we want to show that R+) and R) are linearly independent solutions to the radial equation, which is a second-ordering linear ODE (SOLDE). If we have found two linearly independent solutions, then we know the linear combinations of them build the most general solution. To proceed, first calculate the Wronsklan for the two solutions R+) and R): W()= x Be sure to enter your answer in terms of only X and r. Next, we will calculate the Wronsklan functional (see tutorial section SOLDE, Eqn 5). Using the differential equation for R(r) above, fill in the blanks below for both the function P(r Prime) and the full Wronsklan functional P [P] = exp(- P(rPrime)dr Prime) = exp(- dr Prime)-VI-+-4X 2 x Enter your answers in terms of only ,r and rPrime.Note we are using the dummy variable Prime, written out, rather than your tutorial's r', simply for WebAssign input purposes.
dra -AR=0, where A is a separation constant and R(r) is the radial factor in the solution. + In exercise LP 3, you found that R) and R), with (LP eqn 8) √1 + 4X 1 1/₂ = -3± were both solutions of the radial ODE. In this problem, we want to show that R+) and R) are linearly independent solutions to the radial equation, which is a second-ordering linear ODE (SOLDE). If we have found two linearly independent solutions, then we know the linear combinations of them build the most general solution. To proceed, first calculate the Wronsklan for the two solutions R+) and R): W()= x Be sure to enter your answer in terms of only X and r. Next, we will calculate the Wronsklan functional (see tutorial section SOLDE, Eqn 5). Using the differential equation for R(r) above, fill in the blanks below for both the function P(r Prime) and the full Wronsklan functional P [P] = exp(- P(rPrime)dr Prime) = exp(- dr Prime)-VI-+-4X 2 x Enter your answers in terms of only ,r and rPrime.Note we are using the dummy variable Prime, written out, rather than your tutorial's r', simply for WebAssign input purposes.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![2² +2rd-AR-
-AR-0,
where A is a separation constant and R(r) is the radial factor in the solution.
In exercise LP 3, you found that
R+)
and R), with (LP eqn 8)
√1+4X
1
2-
were both solutions of the radial ODE.
P+ =
In this problem, we want to show that R(+) and R(-) are linearly independent solutions to the radial equation, which is a second-ordering linear ODE (SOLDE). If we have found two linearly independent solutions, then we know the linear combinations of them build the
most general solution.
To proceed, first calculate the Wronskian for the two solutions R(+) and R):
x
Be sure to enter your answer in terms of only A and r.
Next, we will calculate the Wronsklan functional (see tutorial section SOLDE, Eqn 5). Using the differential equation for R(r) above, fill in the blanks below for both the function P(r Prime) and the full Wronskian functional [P].
4[P] = exp(- P(rPrime)dr Prime) = exp(
dr Prime)=√1+4).
,2
x
Enter your answers in terms of only , r and rPrime.Note we are using the dummy variable r Prime, written out, rather than your tutorial's r', simply for WebAssign input purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a731241-1090-4941-b7fa-ade0da10795e%2F3416f6ad-cf84-424d-bde8-98343e7629f6%2Fgx4hu7e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2² +2rd-AR-
-AR-0,
where A is a separation constant and R(r) is the radial factor in the solution.
In exercise LP 3, you found that
R+)
and R), with (LP eqn 8)
√1+4X
1
2-
were both solutions of the radial ODE.
P+ =
In this problem, we want to show that R(+) and R(-) are linearly independent solutions to the radial equation, which is a second-ordering linear ODE (SOLDE). If we have found two linearly independent solutions, then we know the linear combinations of them build the
most general solution.
To proceed, first calculate the Wronskian for the two solutions R(+) and R):
x
Be sure to enter your answer in terms of only A and r.
Next, we will calculate the Wronsklan functional (see tutorial section SOLDE, Eqn 5). Using the differential equation for R(r) above, fill in the blanks below for both the function P(r Prime) and the full Wronskian functional [P].
4[P] = exp(- P(rPrime)dr Prime) = exp(
dr Prime)=√1+4).
,2
x
Enter your answers in terms of only , r and rPrime.Note we are using the dummy variable r Prime, written out, rather than your tutorial's r', simply for WebAssign input purposes.
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