How to solve this Boundary Value problem? [closed]
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$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$
$$u(0)=0$$
$$u(L)=0$$
ordinary-differential-equations boundary-value-problem
$endgroup$
closed as off-topic by Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra Dec 4 '18 at 13:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra
If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$
$$u(0)=0$$
$$u(L)=0$$
ordinary-differential-equations boundary-value-problem
$endgroup$
closed as off-topic by Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra Dec 4 '18 at 13:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra
If this question can be reworded to fit the rules in the help center, please edit the question.
1
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What have you tried? We're not here to do your homework for you.
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– Mattos
Dec 3 '18 at 3:17
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I have tried taking $$u=e^{rx}$$ to find the characteristic equation but I'm not quite sure if that is the way to go
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– anik faisal
Dec 3 '18 at 3:20
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It's not, because you have non-constant coefficients. How about setting $u' = v$ and then using an integrating factor.
$endgroup$
– Mattos
Dec 3 '18 at 3:21
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thanks! I will give it a go
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– anik faisal
Dec 3 '18 at 3:22
2
$begingroup$
@mattos I don't think it would. There's a $u$ term in that equation
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– Dylan
Dec 3 '18 at 13:24
|
show 4 more comments
$begingroup$
$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$
$$u(0)=0$$
$$u(L)=0$$
ordinary-differential-equations boundary-value-problem
$endgroup$
$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$
$$u(0)=0$$
$$u(L)=0$$
ordinary-differential-equations boundary-value-problem
ordinary-differential-equations boundary-value-problem
edited Dec 3 '18 at 18:47
anik faisal
asked Dec 3 '18 at 3:11
anik faisalanik faisal
196
196
closed as off-topic by Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra Dec 4 '18 at 13:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra Dec 4 '18 at 13:06
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Morgan Rodgers, Saad, José Carlos Santos, Cesareo, Vidyanshu Mishra
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
What have you tried? We're not here to do your homework for you.
$endgroup$
– Mattos
Dec 3 '18 at 3:17
$begingroup$
I have tried taking $$u=e^{rx}$$ to find the characteristic equation but I'm not quite sure if that is the way to go
$endgroup$
– anik faisal
Dec 3 '18 at 3:20
$begingroup$
It's not, because you have non-constant coefficients. How about setting $u' = v$ and then using an integrating factor.
$endgroup$
– Mattos
Dec 3 '18 at 3:21
$begingroup$
thanks! I will give it a go
$endgroup$
– anik faisal
Dec 3 '18 at 3:22
2
$begingroup$
@mattos I don't think it would. There's a $u$ term in that equation
$endgroup$
– Dylan
Dec 3 '18 at 13:24
|
show 4 more comments
1
$begingroup$
What have you tried? We're not here to do your homework for you.
$endgroup$
– Mattos
Dec 3 '18 at 3:17
$begingroup$
I have tried taking $$u=e^{rx}$$ to find the characteristic equation but I'm not quite sure if that is the way to go
$endgroup$
– anik faisal
Dec 3 '18 at 3:20
$begingroup$
It's not, because you have non-constant coefficients. How about setting $u' = v$ and then using an integrating factor.
$endgroup$
– Mattos
Dec 3 '18 at 3:21
$begingroup$
thanks! I will give it a go
$endgroup$
– anik faisal
Dec 3 '18 at 3:22
2
$begingroup$
@mattos I don't think it would. There's a $u$ term in that equation
$endgroup$
– Dylan
Dec 3 '18 at 13:24
1
1
$begingroup$
What have you tried? We're not here to do your homework for you.
$endgroup$
– Mattos
Dec 3 '18 at 3:17
$begingroup$
What have you tried? We're not here to do your homework for you.
$endgroup$
– Mattos
Dec 3 '18 at 3:17
$begingroup$
I have tried taking $$u=e^{rx}$$ to find the characteristic equation but I'm not quite sure if that is the way to go
$endgroup$
– anik faisal
Dec 3 '18 at 3:20
$begingroup$
I have tried taking $$u=e^{rx}$$ to find the characteristic equation but I'm not quite sure if that is the way to go
$endgroup$
– anik faisal
Dec 3 '18 at 3:20
$begingroup$
It's not, because you have non-constant coefficients. How about setting $u' = v$ and then using an integrating factor.
$endgroup$
– Mattos
Dec 3 '18 at 3:21
$begingroup$
It's not, because you have non-constant coefficients. How about setting $u' = v$ and then using an integrating factor.
$endgroup$
– Mattos
Dec 3 '18 at 3:21
$begingroup$
thanks! I will give it a go
$endgroup$
– anik faisal
Dec 3 '18 at 3:22
$begingroup$
thanks! I will give it a go
$endgroup$
– anik faisal
Dec 3 '18 at 3:22
2
2
$begingroup$
@mattos I don't think it would. There's a $u$ term in that equation
$endgroup$
– Dylan
Dec 3 '18 at 13:24
$begingroup$
@mattos I don't think it would. There's a $u$ term in that equation
$endgroup$
– Dylan
Dec 3 '18 at 13:24
|
show 4 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Rewrite
$$ (x+x_0)^2frac{d^2u}{dx^2} + 2(x+x_0)frac{du}{dx} - frac{lambda}{k}(x+x_0)^2u = 0 $$
A non-trivial solution only exists for $lambda < 0$. The substitution $z = sqrt{frac{-lambda}{k}}(x+x_0)$ gives
$$ frac{d^2u}{dz^2} + 2zfrac{du}{dz} + z^2 = 0 $$
The solution is in the form of spherical Bessel functions for $n=0$ which happen to have a closed form
$$ u(z) = c_1j_0(z) + c_2y_0(z) = frac{c_1sin z - c_2cos z}{z} $$
The boundary conditions are $u(alpha x_0) = u(alpha (L+x_0)) = 0$ where $alpha = sqrt{frac{-lambda}{k}}$. We need to pick $alpha$ such that two of the zeroes of this function align with the boundary.
Case 1: If $c_2=0$, then the zeroes are the zeroes of $sin z$, which are $z_n = npi$. Since any 2 roots always differ by an integer multiple of $pi$, we have
$$ alpha(L+x_0) - alpha x_0 = alpha L = m pi implies alpha = frac{mpi}{L} $$
where $m$ is an integer. Note that $alpha x_0 = frac{mpi x_0}{L}$ must also be an integer multiple of $pi$, therefore $frac{x_0}{L}$ must be rational.
Case 2: If $c_1=0$, then the zeroes are the zeroes of $cos z$, which are $z_n = (n+frac12)pi$. Similarly, these roots also differ by an integer multiple of $pi$, so we have the same eigenvalue as Case 1, with the condition
$$ alpha x_0 = frac{mpi x_0}{L} = (2n+1)frac{pi}{2} $$
or $frac{2x_0}{L} = frac{2n+1}{m}$ must be rational.
Case 3: If $c_1,c_2 ne 0$, then we can write
$$ u(z) = frac{c_1(sin z - mu cos z)}{z} $$
where the zeroes occur at $tan z_n = mu$ for some $mu in (0,infty)$. These roots also differ by an integer multiple of $pi$, so yet again we have $alpha = frac{mpi}{L}$.
Combining the 3 cases, the eigenfunction can be simplified as
$$ u_m(z) = frac{cos z_m sin z - sin z_m cos z}{z} $$
where $z_m = alpha_m x_0 = frac{mpi x_0}{L}$
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Rewrite
$$ (x+x_0)^2frac{d^2u}{dx^2} + 2(x+x_0)frac{du}{dx} - frac{lambda}{k}(x+x_0)^2u = 0 $$
A non-trivial solution only exists for $lambda < 0$. The substitution $z = sqrt{frac{-lambda}{k}}(x+x_0)$ gives
$$ frac{d^2u}{dz^2} + 2zfrac{du}{dz} + z^2 = 0 $$
The solution is in the form of spherical Bessel functions for $n=0$ which happen to have a closed form
$$ u(z) = c_1j_0(z) + c_2y_0(z) = frac{c_1sin z - c_2cos z}{z} $$
The boundary conditions are $u(alpha x_0) = u(alpha (L+x_0)) = 0$ where $alpha = sqrt{frac{-lambda}{k}}$. We need to pick $alpha$ such that two of the zeroes of this function align with the boundary.
Case 1: If $c_2=0$, then the zeroes are the zeroes of $sin z$, which are $z_n = npi$. Since any 2 roots always differ by an integer multiple of $pi$, we have
$$ alpha(L+x_0) - alpha x_0 = alpha L = m pi implies alpha = frac{mpi}{L} $$
where $m$ is an integer. Note that $alpha x_0 = frac{mpi x_0}{L}$ must also be an integer multiple of $pi$, therefore $frac{x_0}{L}$ must be rational.
Case 2: If $c_1=0$, then the zeroes are the zeroes of $cos z$, which are $z_n = (n+frac12)pi$. Similarly, these roots also differ by an integer multiple of $pi$, so we have the same eigenvalue as Case 1, with the condition
$$ alpha x_0 = frac{mpi x_0}{L} = (2n+1)frac{pi}{2} $$
or $frac{2x_0}{L} = frac{2n+1}{m}$ must be rational.
Case 3: If $c_1,c_2 ne 0$, then we can write
$$ u(z) = frac{c_1(sin z - mu cos z)}{z} $$
where the zeroes occur at $tan z_n = mu$ for some $mu in (0,infty)$. These roots also differ by an integer multiple of $pi$, so yet again we have $alpha = frac{mpi}{L}$.
Combining the 3 cases, the eigenfunction can be simplified as
$$ u_m(z) = frac{cos z_m sin z - sin z_m cos z}{z} $$
where $z_m = alpha_m x_0 = frac{mpi x_0}{L}$
$endgroup$
add a comment |
$begingroup$
Rewrite
$$ (x+x_0)^2frac{d^2u}{dx^2} + 2(x+x_0)frac{du}{dx} - frac{lambda}{k}(x+x_0)^2u = 0 $$
A non-trivial solution only exists for $lambda < 0$. The substitution $z = sqrt{frac{-lambda}{k}}(x+x_0)$ gives
$$ frac{d^2u}{dz^2} + 2zfrac{du}{dz} + z^2 = 0 $$
The solution is in the form of spherical Bessel functions for $n=0$ which happen to have a closed form
$$ u(z) = c_1j_0(z) + c_2y_0(z) = frac{c_1sin z - c_2cos z}{z} $$
The boundary conditions are $u(alpha x_0) = u(alpha (L+x_0)) = 0$ where $alpha = sqrt{frac{-lambda}{k}}$. We need to pick $alpha$ such that two of the zeroes of this function align with the boundary.
Case 1: If $c_2=0$, then the zeroes are the zeroes of $sin z$, which are $z_n = npi$. Since any 2 roots always differ by an integer multiple of $pi$, we have
$$ alpha(L+x_0) - alpha x_0 = alpha L = m pi implies alpha = frac{mpi}{L} $$
where $m$ is an integer. Note that $alpha x_0 = frac{mpi x_0}{L}$ must also be an integer multiple of $pi$, therefore $frac{x_0}{L}$ must be rational.
Case 2: If $c_1=0$, then the zeroes are the zeroes of $cos z$, which are $z_n = (n+frac12)pi$. Similarly, these roots also differ by an integer multiple of $pi$, so we have the same eigenvalue as Case 1, with the condition
$$ alpha x_0 = frac{mpi x_0}{L} = (2n+1)frac{pi}{2} $$
or $frac{2x_0}{L} = frac{2n+1}{m}$ must be rational.
Case 3: If $c_1,c_2 ne 0$, then we can write
$$ u(z) = frac{c_1(sin z - mu cos z)}{z} $$
where the zeroes occur at $tan z_n = mu$ for some $mu in (0,infty)$. These roots also differ by an integer multiple of $pi$, so yet again we have $alpha = frac{mpi}{L}$.
Combining the 3 cases, the eigenfunction can be simplified as
$$ u_m(z) = frac{cos z_m sin z - sin z_m cos z}{z} $$
where $z_m = alpha_m x_0 = frac{mpi x_0}{L}$
$endgroup$
add a comment |
$begingroup$
Rewrite
$$ (x+x_0)^2frac{d^2u}{dx^2} + 2(x+x_0)frac{du}{dx} - frac{lambda}{k}(x+x_0)^2u = 0 $$
A non-trivial solution only exists for $lambda < 0$. The substitution $z = sqrt{frac{-lambda}{k}}(x+x_0)$ gives
$$ frac{d^2u}{dz^2} + 2zfrac{du}{dz} + z^2 = 0 $$
The solution is in the form of spherical Bessel functions for $n=0$ which happen to have a closed form
$$ u(z) = c_1j_0(z) + c_2y_0(z) = frac{c_1sin z - c_2cos z}{z} $$
The boundary conditions are $u(alpha x_0) = u(alpha (L+x_0)) = 0$ where $alpha = sqrt{frac{-lambda}{k}}$. We need to pick $alpha$ such that two of the zeroes of this function align with the boundary.
Case 1: If $c_2=0$, then the zeroes are the zeroes of $sin z$, which are $z_n = npi$. Since any 2 roots always differ by an integer multiple of $pi$, we have
$$ alpha(L+x_0) - alpha x_0 = alpha L = m pi implies alpha = frac{mpi}{L} $$
where $m$ is an integer. Note that $alpha x_0 = frac{mpi x_0}{L}$ must also be an integer multiple of $pi$, therefore $frac{x_0}{L}$ must be rational.
Case 2: If $c_1=0$, then the zeroes are the zeroes of $cos z$, which are $z_n = (n+frac12)pi$. Similarly, these roots also differ by an integer multiple of $pi$, so we have the same eigenvalue as Case 1, with the condition
$$ alpha x_0 = frac{mpi x_0}{L} = (2n+1)frac{pi}{2} $$
or $frac{2x_0}{L} = frac{2n+1}{m}$ must be rational.
Case 3: If $c_1,c_2 ne 0$, then we can write
$$ u(z) = frac{c_1(sin z - mu cos z)}{z} $$
where the zeroes occur at $tan z_n = mu$ for some $mu in (0,infty)$. These roots also differ by an integer multiple of $pi$, so yet again we have $alpha = frac{mpi}{L}$.
Combining the 3 cases, the eigenfunction can be simplified as
$$ u_m(z) = frac{cos z_m sin z - sin z_m cos z}{z} $$
where $z_m = alpha_m x_0 = frac{mpi x_0}{L}$
$endgroup$
Rewrite
$$ (x+x_0)^2frac{d^2u}{dx^2} + 2(x+x_0)frac{du}{dx} - frac{lambda}{k}(x+x_0)^2u = 0 $$
A non-trivial solution only exists for $lambda < 0$. The substitution $z = sqrt{frac{-lambda}{k}}(x+x_0)$ gives
$$ frac{d^2u}{dz^2} + 2zfrac{du}{dz} + z^2 = 0 $$
The solution is in the form of spherical Bessel functions for $n=0$ which happen to have a closed form
$$ u(z) = c_1j_0(z) + c_2y_0(z) = frac{c_1sin z - c_2cos z}{z} $$
The boundary conditions are $u(alpha x_0) = u(alpha (L+x_0)) = 0$ where $alpha = sqrt{frac{-lambda}{k}}$. We need to pick $alpha$ such that two of the zeroes of this function align with the boundary.
Case 1: If $c_2=0$, then the zeroes are the zeroes of $sin z$, which are $z_n = npi$. Since any 2 roots always differ by an integer multiple of $pi$, we have
$$ alpha(L+x_0) - alpha x_0 = alpha L = m pi implies alpha = frac{mpi}{L} $$
where $m$ is an integer. Note that $alpha x_0 = frac{mpi x_0}{L}$ must also be an integer multiple of $pi$, therefore $frac{x_0}{L}$ must be rational.
Case 2: If $c_1=0$, then the zeroes are the zeroes of $cos z$, which are $z_n = (n+frac12)pi$. Similarly, these roots also differ by an integer multiple of $pi$, so we have the same eigenvalue as Case 1, with the condition
$$ alpha x_0 = frac{mpi x_0}{L} = (2n+1)frac{pi}{2} $$
or $frac{2x_0}{L} = frac{2n+1}{m}$ must be rational.
Case 3: If $c_1,c_2 ne 0$, then we can write
$$ u(z) = frac{c_1(sin z - mu cos z)}{z} $$
where the zeroes occur at $tan z_n = mu$ for some $mu in (0,infty)$. These roots also differ by an integer multiple of $pi$, so yet again we have $alpha = frac{mpi}{L}$.
Combining the 3 cases, the eigenfunction can be simplified as
$$ u_m(z) = frac{cos z_m sin z - sin z_m cos z}{z} $$
where $z_m = alpha_m x_0 = frac{mpi x_0}{L}$
edited Dec 13 '18 at 5:48
answered Dec 4 '18 at 9:16
DylanDylan
12.9k31027
12.9k31027
add a comment |
add a comment |
1
$begingroup$
What have you tried? We're not here to do your homework for you.
$endgroup$
– Mattos
Dec 3 '18 at 3:17
$begingroup$
I have tried taking $$u=e^{rx}$$ to find the characteristic equation but I'm not quite sure if that is the way to go
$endgroup$
– anik faisal
Dec 3 '18 at 3:20
$begingroup$
It's not, because you have non-constant coefficients. How about setting $u' = v$ and then using an integrating factor.
$endgroup$
– Mattos
Dec 3 '18 at 3:21
$begingroup$
thanks! I will give it a go
$endgroup$
– anik faisal
Dec 3 '18 at 3:22
2
$begingroup$
@mattos I don't think it would. There's a $u$ term in that equation
$endgroup$
– Dylan
Dec 3 '18 at 13:24