How to solve this Boundary Value problem? [closed]












-3












$begingroup$


$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$



$$u(0)=0$$
$$u(L)=0$$










share|cite|improve this question











$endgroup$



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




    $begingroup$
    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
















-3












$begingroup$


$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$



$$u(0)=0$$
$$u(L)=0$$










share|cite|improve this question











$endgroup$



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




    $begingroup$
    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














-3












-3








-3


2



$begingroup$


$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$



$$u(0)=0$$
$$u(L)=0$$










share|cite|improve this question











$endgroup$




$$ left(frac{d^2 u}{d x^2} right)+frac{2}{x+x_0} frac{d u}{d x}=frac{lambda}{k} u; quad x in [0,L], k,x_0>0 $$



$$u(0)=0$$
$$u(L)=0$$







ordinary-differential-equations boundary-value-problem






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








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














  • 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










1 Answer
1






active

oldest

votes


















1












$begingroup$

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 $z_m = alpha_m x_0 = frac{mpi x_0}{L}$






share|cite|improve this answer











$endgroup$




















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    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 $z_m = alpha_m x_0 = frac{mpi x_0}{L}$






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      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 $z_m = alpha_m x_0 = frac{mpi x_0}{L}$






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        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 $z_m = alpha_m x_0 = frac{mpi x_0}{L}$






        share|cite|improve this answer











        $endgroup$



        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 $z_m = alpha_m x_0 = frac{mpi x_0}{L}$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 13 '18 at 5:48

























        answered Dec 4 '18 at 9:16









        DylanDylan

        12.9k31027




        12.9k31027















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