Behaviour of Solutions to $x^2y'' + alpha xy'+ beta y = 0$ as $x to 0$ and $x to infty$Find $alpha$ such that $y'=sqrt1+y^4-|y|^alpha$ has global solutionsWhy do we use $x^m_1$ instead of $e^m_1x$ for general solutions to Cauchy-Euler equations?When all solutions of $y''+ay'+by=0$ are bounded in R?Find all values of $alpha$ so that all solutions approach $0$ as $x to infty$Behaviour of solutions to ODE near singular pointsFind the value $alpha$ s.t. all solutions tend to 0 as $ttoinfty$Closed form for $intfracleft((x + i) betaright)^beta x^beta - 2(x^2 + 1)^beta expleft(-fracalphaA xright) , mathrmd x$Indicial equation of $(x^2-1)^2y''+(x+1)y'-y=0$Solutions for Euler EquationsSolutions of the Falkner Skan Equation for Negative Beta
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Behaviour of Solutions to $x^2y'' + alpha xy'+ beta y = 0$ as $x to 0$ and $x to infty$
Find $alpha$ such that $y'=sqrt1+y^4-|y|^alpha$ has global solutionsWhy do we use $x^m_1$ instead of $e^m_1x$ for general solutions to Cauchy-Euler equations?When all solutions of $y''+ay'+by=0$ are bounded in R?Find all values of $alpha$ so that all solutions approach $0$ as $x to infty$Behaviour of solutions to ODE near singular pointsFind the value $alpha$ s.t. all solutions tend to 0 as $ttoinfty$Closed form for $intfracleft((x + i) betaright)^beta x^beta - 2(x^2 + 1)^beta expleft(-fracalphaA xright) , mathrmd x$Indicial equation of $(x^2-1)^2y''+(x+1)y'-y=0$Solutions for Euler EquationsSolutions of the Falkner Skan Equation for Negative Beta
$begingroup$
Consider the Euler equation $x^2y'' + alpha xy' + beta y = 0$.
Find conditions on $alpha$ and $beta$ so that:
- All solutions approach zero as $x rightarrow 0$
- All solutions are bounded as $x rightarrow 0$
- All solutions approach zero as $x rightarrow infty$
- All solutions are bounded as $x rightarrow infty$
- All solutions are bounded both as $x rightarrow 0$ and $x rightarrow infty$
I am having trouble solving the problem. It must obviously have to do with the solutions to the indicial equation $ r^2 + (alpha - 1)r + beta= 0 implies r = dfrac1 - alpha
pm sqrt (alpha - 1)^2 - 4beta^22 $. When these roots are identical or complex the solutions involve an $ln x$ term the limit of which when $x to 0$ is undefined I think? So my feeble argument is that the solutions for $r$ should be distinct to begin with? Please help.
ordinary-differential-equations
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
$endgroup$
add a comment |
$begingroup$
Consider the Euler equation $x^2y'' + alpha xy' + beta y = 0$.
Find conditions on $alpha$ and $beta$ so that:
- All solutions approach zero as $x rightarrow 0$
- All solutions are bounded as $x rightarrow 0$
- All solutions approach zero as $x rightarrow infty$
- All solutions are bounded as $x rightarrow infty$
- All solutions are bounded both as $x rightarrow 0$ and $x rightarrow infty$
I am having trouble solving the problem. It must obviously have to do with the solutions to the indicial equation $ r^2 + (alpha - 1)r + beta= 0 implies r = dfrac1 - alpha
pm sqrt (alpha - 1)^2 - 4beta^22 $. When these roots are identical or complex the solutions involve an $ln x$ term the limit of which when $x to 0$ is undefined I think? So my feeble argument is that the solutions for $r$ should be distinct to begin with? Please help.
ordinary-differential-equations
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
$endgroup$
add a comment |
$begingroup$
Consider the Euler equation $x^2y'' + alpha xy' + beta y = 0$.
Find conditions on $alpha$ and $beta$ so that:
- All solutions approach zero as $x rightarrow 0$
- All solutions are bounded as $x rightarrow 0$
- All solutions approach zero as $x rightarrow infty$
- All solutions are bounded as $x rightarrow infty$
- All solutions are bounded both as $x rightarrow 0$ and $x rightarrow infty$
I am having trouble solving the problem. It must obviously have to do with the solutions to the indicial equation $ r^2 + (alpha - 1)r + beta= 0 implies r = dfrac1 - alpha
pm sqrt (alpha - 1)^2 - 4beta^22 $. When these roots are identical or complex the solutions involve an $ln x$ term the limit of which when $x to 0$ is undefined I think? So my feeble argument is that the solutions for $r$ should be distinct to begin with? Please help.
ordinary-differential-equations
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
$endgroup$
Consider the Euler equation $x^2y'' + alpha xy' + beta y = 0$.
Find conditions on $alpha$ and $beta$ so that:
- All solutions approach zero as $x rightarrow 0$
- All solutions are bounded as $x rightarrow 0$
- All solutions approach zero as $x rightarrow infty$
- All solutions are bounded as $x rightarrow infty$
- All solutions are bounded both as $x rightarrow 0$ and $x rightarrow infty$
I am having trouble solving the problem. It must obviously have to do with the solutions to the indicial equation $ r^2 + (alpha - 1)r + beta= 0 implies r = dfrac1 - alpha
pm sqrt (alpha - 1)^2 - 4beta^22 $. When these roots are identical or complex the solutions involve an $ln x$ term the limit of which when $x to 0$ is undefined I think? So my feeble argument is that the solutions for $r$ should be distinct to begin with? Please help.
ordinary-differential-equations
ordinary-differential-equations
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
asked May 1 '14 at 6:39
IshfaaqIshfaaq
7,95811443
7,95811443
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
A logarithmic term, if present, only changes the behaviour as $x to 0$ or $x to infty$ when $textRe(r) = 0$. That is, if $textRe(r) ne 0$ then both
$x^r$ and $x^r log(x)$ have the same limit or lack of limit as $x to infty$ and as $x to 0$. I don't know what that "or complex" is doing there: in the Euler equation a logarithmic term only appears when the roots are equal.
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
It's literally worked out here:
http://banach.millersville.edu/~bob/math365/homework/boycediprima/sec05.05.pdf
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
$endgroup$
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
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votes
active
oldest
votes
$begingroup$
A logarithmic term, if present, only changes the behaviour as $x to 0$ or $x to infty$ when $textRe(r) = 0$. That is, if $textRe(r) ne 0$ then both
$x^r$ and $x^r log(x)$ have the same limit or lack of limit as $x to infty$ and as $x to 0$. I don't know what that "or complex" is doing there: in the Euler equation a logarithmic term only appears when the roots are equal.
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
A logarithmic term, if present, only changes the behaviour as $x to 0$ or $x to infty$ when $textRe(r) = 0$. That is, if $textRe(r) ne 0$ then both
$x^r$ and $x^r log(x)$ have the same limit or lack of limit as $x to infty$ and as $x to 0$. I don't know what that "or complex" is doing there: in the Euler equation a logarithmic term only appears when the roots are equal.
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
A logarithmic term, if present, only changes the behaviour as $x to 0$ or $x to infty$ when $textRe(r) = 0$. That is, if $textRe(r) ne 0$ then both
$x^r$ and $x^r log(x)$ have the same limit or lack of limit as $x to infty$ and as $x to 0$. I don't know what that "or complex" is doing there: in the Euler equation a logarithmic term only appears when the roots are equal.
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
A logarithmic term, if present, only changes the behaviour as $x to 0$ or $x to infty$ when $textRe(r) = 0$. That is, if $textRe(r) ne 0$ then both
$x^r$ and $x^r log(x)$ have the same limit or lack of limit as $x to infty$ and as $x to 0$. I don't know what that "or complex" is doing there: in the Euler equation a logarithmic term only appears when the roots are equal.
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
answered May 1 '14 at 7:09
Robert IsraelRobert Israel
330k23219473
330k23219473
add a comment |
add a comment |
$begingroup$
It's literally worked out here:
http://banach.millersville.edu/~bob/math365/homework/boycediprima/sec05.05.pdf
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
$endgroup$
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
add a comment |
$begingroup$
It's literally worked out here:
http://banach.millersville.edu/~bob/math365/homework/boycediprima/sec05.05.pdf
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
$endgroup$
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
add a comment |
$begingroup$
It's literally worked out here:
http://banach.millersville.edu/~bob/math365/homework/boycediprima/sec05.05.pdf
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
$endgroup$
It's literally worked out here:
http://banach.millersville.edu/~bob/math365/homework/boycediprima/sec05.05.pdf
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
answered Mar 20 at 19:33
Wesley StrikWesley Strik
2,189424
2,189424
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
add a comment |
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
$begingroup$
Note this is question 5.4.29 from Boyce's elementary diff eqns version 11 global edition - exercise 27 in the American edition.
$endgroup$
– Wesley Strik
Mar 20 at 19:33
add a comment |
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