Solution of the nonlinear ODE $y'' y' =A y' y + B (x-1) y$, with $y(0) = 1$, $y(1) = 0$

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What analytical techniques are available for finding solutions to the nonlinear ODE
$$y'' y' =A y' y + B (x-1) y,$$ with boundary conditions $$y(0) = 1, quad y(1) = 0,$$
where $A$ and $B$ are positive, real constants? Unfortunately, neither $A$ nor $B$ are necessarily small.

Does assuming that $y'(x) neq 0$ allow further progress?

differential-equations

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asked Jul 25 at 15:12

mjr

634

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Note that $$y(x)=0$$ is one solution.
  – Dr. Sonnhard Graubner
  Jul 25 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $y(x) = 0$ does not satisfy the boundary condition $y(0) = 1$
  – mjr
  Jul 25 at 15:41
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Did you try the Bellman-Kalaba Quasi-linearization method?
  – Cesareo
  Jul 25 at 16:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the idea - I haven't tried this yet. Is rand.org/pubs/reports/R438.html the standard reference for the technique? I cannot access this paper through my university without paying for a copy.
  – mjr
  Jul 25 at 16:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  i can send you a copy if that helps
  – user577488
  Jul 25 at 21:31
  

  
  
  
  

add a comment | 

up vote
4
down vote

favorite

3

What analytical techniques are available for finding solutions to the nonlinear ODE
$$y'' y' =A y' y + B (x-1) y,$$ with boundary conditions $$y(0) = 1, quad y(1) = 0,$$
where $A$ and $B$ are positive, real constants? Unfortunately, neither $A$ nor $B$ are necessarily small.

Does assuming that $y'(x) neq 0$ allow further progress?

differential-equations

share|cite|improve this question

asked Jul 25 at 15:12

mjr

634

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Note that $$y(x)=0$$ is one solution.
  – Dr. Sonnhard Graubner
  Jul 25 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $y(x) = 0$ does not satisfy the boundary condition $y(0) = 1$
  – mjr
  Jul 25 at 15:41
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Did you try the Bellman-Kalaba Quasi-linearization method?
  – Cesareo
  Jul 25 at 16:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the idea - I haven't tried this yet. Is rand.org/pubs/reports/R438.html the standard reference for the technique? I cannot access this paper through my university without paying for a copy.
  – mjr
  Jul 25 at 16:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  i can send you a copy if that helps
  – user577488
  Jul 25 at 21:31
  

  
  
  
  

add a comment | 

up vote
4
down vote

favorite

3

up vote
4
down vote

favorite

3

3

What analytical techniques are available for finding solutions to the nonlinear ODE
$$y'' y' =A y' y + B (x-1) y,$$ with boundary conditions $$y(0) = 1, quad y(1) = 0,$$
where $A$ and $B$ are positive, real constants? Unfortunately, neither $A$ nor $B$ are necessarily small.

Does assuming that $y'(x) neq 0$ allow further progress?

differential-equations

share|cite|improve this question

asked Jul 25 at 15:12

mjr

634

What analytical techniques are available for finding solutions to the nonlinear ODE
$$y'' y' =A y' y + B (x-1) y,$$ with boundary conditions $$y(0) = 1, quad y(1) = 0,$$
where $A$ and $B$ are positive, real constants? Unfortunately, neither $A$ nor $B$ are necessarily small.

Does assuming that $y'(x) neq 0$ allow further progress?

differential-equations

share|cite|improve this question

asked Jul 25 at 15:12

mjr

634

share|cite|improve this question

share|cite|improve this question

asked Jul 25 at 15:12

mjr

634

asked Jul 25 at 15:12

mjr

634

asked Jul 25 at 15:12

mjr

634

634

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Note that $$y(x)=0$$ is one solution.
  – Dr. Sonnhard Graubner
  Jul 25 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $y(x) = 0$ does not satisfy the boundary condition $y(0) = 1$
  – mjr
  Jul 25 at 15:41
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Did you try the Bellman-Kalaba Quasi-linearization method?
  – Cesareo
  Jul 25 at 16:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the idea - I haven't tried this yet. Is rand.org/pubs/reports/R438.html the standard reference for the technique? I cannot access this paper through my university without paying for a copy.
  – mjr
  Jul 25 at 16:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  i can send you a copy if that helps
  – user577488
  Jul 25 at 21:31
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Note that $$y(x)=0$$ is one solution.
  – Dr. Sonnhard Graubner
  Jul 25 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $y(x) = 0$ does not satisfy the boundary condition $y(0) = 1$
  – mjr
  Jul 25 at 15:41
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Did you try the Bellman-Kalaba Quasi-linearization method?
  – Cesareo
  Jul 25 at 16:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the idea - I haven't tried this yet. Is rand.org/pubs/reports/R438.html the standard reference for the technique? I cannot access this paper through my university without paying for a copy.
  – mjr
  Jul 25 at 16:55
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  i can send you a copy if that helps
  – user577488
  Jul 25 at 21:31
  

  
  
  
  

Note that $$y(x)=0$$ is one solution.
– Dr. Sonnhard Graubner
Jul 25 at 15:35

Note that $$y(x)=0$$ is one solution.
– Dr. Sonnhard Graubner
Jul 25 at 15:35

1

1

$y(x) = 0$ does not satisfy the boundary condition $y(0) = 1$
– mjr
Jul 25 at 15:41

$y(x) = 0$ does not satisfy the boundary condition $y(0) = 1$
– mjr
Jul 25 at 15:41

1

1

Did you try the Bellman-Kalaba Quasi-linearization method?
– Cesareo
Jul 25 at 16:09

Did you try the Bellman-Kalaba Quasi-linearization method?
– Cesareo
Jul 25 at 16:09

Thanks for the idea - I haven't tried this yet. Is rand.org/pubs/reports/R438.html the standard reference for the technique? I cannot access this paper through my university without paying for a copy.
– mjr
Jul 25 at 16:55

Thanks for the idea - I haven't tried this yet. Is rand.org/pubs/reports/R438.html the standard reference for the technique? I cannot access this paper through my university without paying for a copy.
– mjr
Jul 25 at 16:55

2

2

i can send you a copy if that helps
– user577488
Jul 25 at 21:31

i can send you a copy if that helps
– user577488
Jul 25 at 21:31

add a comment | 

1 Answer
1

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up vote
0
down vote

Hint:

Let $t=x-1$ ,

Then $y'y''=Ayy'+Bty$ with $y(-1)=1$ and $y(0)=0$

$y''=Ay+dfracBtyy'$

You can consider as the ODE of the type http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=429

share|cite|improve this answer

edited Jul 27 at 13:01

answered Jul 26 at 12:43

doraemonpaul

12k31660

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
  – mjr
  Jul 26 at 13:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
  – mjr
  Jul 27 at 17:16
  
  

  
  
  
  

add a comment | 

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1 Answer
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1 Answer
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up vote
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Hint:

Let $t=x-1$ ,

Then $y'y''=Ayy'+Bty$ with $y(-1)=1$ and $y(0)=0$

$y''=Ay+dfracBtyy'$

You can consider as the ODE of the type http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=429

share|cite|improve this answer

edited Jul 27 at 13:01

answered Jul 26 at 12:43

doraemonpaul

12k31660

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
  – mjr
  Jul 26 at 13:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
  – mjr
  Jul 27 at 17:16
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

Hint:

Let $t=x-1$ ,

Then $y'y''=Ayy'+Bty$ with $y(-1)=1$ and $y(0)=0$

$y''=Ay+dfracBtyy'$

You can consider as the ODE of the type http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=429

share|cite|improve this answer

edited Jul 27 at 13:01

answered Jul 26 at 12:43

doraemonpaul

12k31660

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
  – mjr
  Jul 26 at 13:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
  – mjr
  Jul 27 at 17:16
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

Hint:

Let $t=x-1$ ,

Then $y'y''=Ayy'+Bty$ with $y(-1)=1$ and $y(0)=0$

$y''=Ay+dfracBtyy'$

You can consider as the ODE of the type http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=429

share|cite|improve this answer

edited Jul 27 at 13:01

answered Jul 26 at 12:43

doraemonpaul

12k31660

Hint:

Let $t=x-1$ ,

Then $y'y''=Ayy'+Bty$ with $y(-1)=1$ and $y(0)=0$

$y''=Ay+dfracBtyy'$

You can consider as the ODE of the type http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=429

share|cite|improve this answer

edited Jul 27 at 13:01

answered Jul 26 at 12:43

doraemonpaul

12k31660

share|cite|improve this answer

share|cite|improve this answer

edited Jul 27 at 13:01

edited Jul 27 at 13:01

edited Jul 27 at 13:01

answered Jul 26 at 12:43

doraemonpaul

12k31660

answered Jul 26 at 12:43

doraemonpaul

12k31660

answered Jul 26 at 12:43

doraemonpaul

12k31660

12k31660

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
  – mjr
  Jul 26 at 13:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
  – mjr
  Jul 27 at 17:16
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
  – mjr
  Jul 26 at 13:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
  – mjr
  Jul 27 at 17:16
  
  

  
  
  
  

1

1

Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
– mjr
Jul 26 at 13:44

Thanks. Do you have a further hint? This substitution for $x$ doesn't seem to have much effect on the form of the equation. I've been looking for substitutions for $y$ without much luck so far
– mjr
Jul 26 at 13:44

It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
– mjr
Jul 27 at 17:16

It would be great to have some more details. I found the book you linked to, although somewhere else because the link seems broken, and unfortunately the section (2.6, and specifically 2.6.4) that discusses equations of the relevant form doesn't appear to cover the actual combination of terms in the question - unless I missed it! Do you know if it's covered in the book? Thanks
– mjr
Jul 27 at 17:16

add a comment | 

 

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