Find the solution of a distributional differential equation [duplicate]
$begingroup$
This question already has an answer here:
General distributional solution of the Airy Equation
1 answer
How can I find the solution of the distributional equation
$$ frac{d^2u}{dx^2}-xu = 0 $$
And prove that there are at least two linear independent solutions? I've tried to to with the Fourier Transform but it is not possible to find the solution that diverges as $x$ tends to infinity, because the Fourier Transform is not defined for such functions. (they are so called Airy functions). I've tried also to solve it by testing the equation with an appropriate test function, but I'm stuck. 
This is the method used. Once I've found Ai(x), I cannot prove that there are two independent solutions.
differential-equations distribution-theory airy-functions
asked Jan 6 at 23:48
MargeAMargeA
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marked as duplicate by LutzL differential-equations
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Jan 7 at 9:50
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
General distributional solution of the Airy Equation
1 answer
How can I find the solution of the distributional equation
$$ frac{d^2u}{dx^2}-xu = 0 $$
And prove that there are at least two linear independent solutions? I've tried to to with the Fourier Transform but it is not possible to find the solution that diverges as $x$ tends to infinity, because the Fourier Transform is not defined for such functions. (they are so called Airy functions). I've tried also to solve it by testing the equation with an appropriate test function, but I'm stuck.
This is the method used. Once I've found Ai(x), I cannot prove that there are two independent solutions.
differential-equations distribution-theory airy-functions
asked Jan 6 at 23:48
MargeAMargeA
162
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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marked as duplicate by LutzL differential-equations
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Jan 7 at 9:50
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
You may try this form $u=sum_{n=-infty}^{infty}a_nx^n$, and seek for the recurrence relation between $a_n$'s.
$endgroup$
– hypernova
Jan 7 at 1:12
$begingroup$
eqworld.ipmnet.ru/en/solutions/ode/ode0202.pdf
$endgroup$
– player100
Jan 7 at 2:00
$begingroup$
Why do you need test functions? Classical solutions work just fine, and you argue like you would for any linear second-order ODE. For example, the solutions with $(u(0),u'(0))$ equal to $(1,0)$ and $(0,1)$, respectively, are linearly independent.
$endgroup$
– Hans Lundmark
Jan 7 at 7:51
$begingroup$
If you do not specify what solution method you use and exactly what the point is you wonder about, you will always get the same answer that you already got for your first identical question, before you added the Fourier calculations.
$endgroup$
– LutzL
Jan 7 at 9:51
add a comment |
$begingroup$
This question already has an answer here:
General distributional solution of the Airy Equation
1 answer
How can I find the solution of the distributional equation
$$ frac{d^2u}{dx^2}-xu = 0 $$
And prove that there are at least two linear independent solutions? I've tried to to with the Fourier Transform but it is not possible to find the solution that diverges as $x$ tends to infinity, because the Fourier Transform is not defined for such functions. (they are so called Airy functions). I've tried also to solve it by testing the equation with an appropriate test function, but I'm stuck.
This is the method used. Once I've found Ai(x), I cannot prove that there are two independent solutions.
differential-equations distribution-theory airy-functions
asked Jan 6 at 23:48
MargeAMargeA
162
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
This question already has an answer here:
General distributional solution of the Airy Equation
1 answer
How can I find the solution of the distributional equation
$$ frac{d^2u}{dx^2}-xu = 0 $$
And prove that there are at least two linear independent solutions? I've tried to to with the Fourier Transform but it is not possible to find the solution that diverges as $x$ tends to infinity, because the Fourier Transform is not defined for such functions. (they are so called Airy functions). I've tried also to solve it by testing the equation with an appropriate test function, but I'm stuck.
This is the method used. Once I've found Ai(x), I cannot prove that there are two independent solutions.
This question already has an answer here:
General distributional solution of the Airy Equation
1 answer
differential-equations distribution-theory airy-functions
differential-equations distribution-theory airy-functions
asked Jan 6 at 23:48
MargeAMargeA
162
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 6 at 23:48
MargeAMargeA
162
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 7 at 9:56
MargeA
asked Jan 6 at 23:48
MargeAMargeA
162
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 6 at 23:48
MargeAMargeA
162
asked Jan 6 at 23:48
MargeAMargeA
162
162
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
MargeA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
marked as duplicate by LutzL differential-equations
Users with the differential-equations badge can single-handedly close differential-equations questions as duplicates and reopen them as needed.
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Jan 7 at 9:50
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by LutzL differential-equations
Users with the differential-equations badge can single-handedly close differential-equations questions as duplicates and reopen them as needed.
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Jan 7 at 9:50
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
You may try this form $u=sum_{n=-infty}^{infty}a_nx^n$, and seek for the recurrence relation between $a_n$'s.
$endgroup$
– hypernova
Jan 7 at 1:12
$begingroup$
eqworld.ipmnet.ru/en/solutions/ode/ode0202.pdf
$endgroup$
– player100
Jan 7 at 2:00
$begingroup$
Why do you need test functions? Classical solutions work just fine, and you argue like you would for any linear second-order ODE. For example, the solutions with $(u(0),u'(0))$ equal to $(1,0)$ and $(0,1)$, respectively, are linearly independent.
$endgroup$
– Hans Lundmark
Jan 7 at 7:51
$begingroup$
If you do not specify what solution method you use and exactly what the point is you wonder about, you will always get the same answer that you already got for your first identical question, before you added the Fourier calculations.
$endgroup$
– LutzL
Jan 7 at 9:51
add a comment |
$begingroup$
You may try this form $u=sum_{n=-infty}^{infty}a_nx^n$, and seek for the recurrence relation between $a_n$'s.
$endgroup$
– hypernova
Jan 7 at 1:12
$begingroup$
eqworld.ipmnet.ru/en/solutions/ode/ode0202.pdf
$endgroup$
– player100
Jan 7 at 2:00
$begingroup$
Why do you need test functions? Classical solutions work just fine, and you argue like you would for any linear second-order ODE. For example, the solutions with $(u(0),u'(0))$ equal to $(1,0)$ and $(0,1)$, respectively, are linearly independent.
$endgroup$
– Hans Lundmark
Jan 7 at 7:51
$begingroup$
If you do not specify what solution method you use and exactly what the point is you wonder about, you will always get the same answer that you already got for your first identical question, before you added the Fourier calculations.
$endgroup$
– LutzL
Jan 7 at 9:51
$begingroup$
You may try this form $u=sum_{n=-infty}^{infty}a_nx^n$, and seek for the recurrence relation between $a_n$'s.
$endgroup$
– hypernova
Jan 7 at 1:12
$begingroup$
You may try this form $u=sum_{n=-infty}^{infty}a_nx^n$, and seek for the recurrence relation between $a_n$'s.
$endgroup$
– hypernova
Jan 7 at 1:12
$begingroup$
eqworld.ipmnet.ru/en/solutions/ode/ode0202.pdf
$endgroup$
– player100
Jan 7 at 2:00
$begingroup$
eqworld.ipmnet.ru/en/solutions/ode/ode0202.pdf
$endgroup$
– player100
Jan 7 at 2:00
$begingroup$
Why do you need test functions? Classical solutions work just fine, and you argue like you would for any linear second-order ODE. For example, the solutions with $(u(0),u'(0))$ equal to $(1,0)$ and $(0,1)$, respectively, are linearly independent.
$endgroup$
– Hans Lundmark
Jan 7 at 7:51
$begingroup$
Why do you need test functions? Classical solutions work just fine, and you argue like you would for any linear second-order ODE. For example, the solutions with $(u(0),u'(0))$ equal to $(1,0)$ and $(0,1)$, respectively, are linearly independent.
$endgroup$
– Hans Lundmark
Jan 7 at 7:51
$begingroup$
If you do not specify what solution method you use and exactly what the point is you wonder about, you will always get the same answer that you already got for your first identical question, before you added the Fourier calculations.
$endgroup$
– LutzL
Jan 7 at 9:51
$begingroup$
If you do not specify what solution method you use and exactly what the point is you wonder about, you will always get the same answer that you already got for your first identical question, before you added the Fourier calculations.
$endgroup$
– LutzL
Jan 7 at 9:51
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
NOTE : The original question has been modified and completed after discussion. As a consequence my answer below is no longer well-adapted.
$$ frac{d^2u}{dx^2}-xu = 0 $$
This ODE is known as Airy equation : http://mathworld.wolfram.com/AiryDifferentialEquation.html
There is no closed form for the solutions with a finite number of elementary functions.
A closed form requires special functions, namely the Airy functions Ai$(x)$ and Bi$(x)$
$$u(x)=c_1text{Ai}(x)+c_2text{Bi}(x)$$
The Airy functions are related to some other special functions, especially the Bessel functions of first kind.
To express the solutions with elementary fonctions only, infinite series are required. The calculus is rather boring. See Eqs.$(2-26)$ in the above reference.
About properties of Airy functions : http://mathworld.wolfram.com/AiryFunctions.html
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
$endgroup$
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
|
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
NOTE : The original question has been modified and completed after discussion. As a consequence my answer below is no longer well-adapted.
$$ frac{d^2u}{dx^2}-xu = 0 $$
This ODE is known as Airy equation : http://mathworld.wolfram.com/AiryDifferentialEquation.html
There is no closed form for the solutions with a finite number of elementary functions.
A closed form requires special functions, namely the Airy functions Ai$(x)$ and Bi$(x)$
$$u(x)=c_1text{Ai}(x)+c_2text{Bi}(x)$$
The Airy functions are related to some other special functions, especially the Bessel functions of first kind.
To express the solutions with elementary fonctions only, infinite series are required. The calculus is rather boring. See Eqs.$(2-26)$ in the above reference.
About properties of Airy functions : http://mathworld.wolfram.com/AiryFunctions.html
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
$endgroup$
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
|
show 2 more comments
$begingroup$
NOTE : The original question has been modified and completed after discussion. As a consequence my answer below is no longer well-adapted.
$$ frac{d^2u}{dx^2}-xu = 0 $$
This ODE is known as Airy equation : http://mathworld.wolfram.com/AiryDifferentialEquation.html
There is no closed form for the solutions with a finite number of elementary functions.
A closed form requires special functions, namely the Airy functions Ai$(x)$ and Bi$(x)$
$$u(x)=c_1text{Ai}(x)+c_2text{Bi}(x)$$
The Airy functions are related to some other special functions, especially the Bessel functions of first kind.
To express the solutions with elementary fonctions only, infinite series are required. The calculus is rather boring. See Eqs.$(2-26)$ in the above reference.
About properties of Airy functions : http://mathworld.wolfram.com/AiryFunctions.html
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
$endgroup$
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
|
show 2 more comments
$begingroup$
NOTE : The original question has been modified and completed after discussion. As a consequence my answer below is no longer well-adapted.
$$ frac{d^2u}{dx^2}-xu = 0 $$
This ODE is known as Airy equation : http://mathworld.wolfram.com/AiryDifferentialEquation.html
There is no closed form for the solutions with a finite number of elementary functions.
A closed form requires special functions, namely the Airy functions Ai$(x)$ and Bi$(x)$
$$u(x)=c_1text{Ai}(x)+c_2text{Bi}(x)$$
The Airy functions are related to some other special functions, especially the Bessel functions of first kind.
To express the solutions with elementary fonctions only, infinite series are required. The calculus is rather boring. See Eqs.$(2-26)$ in the above reference.
About properties of Airy functions : http://mathworld.wolfram.com/AiryFunctions.html
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
$endgroup$
NOTE : The original question has been modified and completed after discussion. As a consequence my answer below is no longer well-adapted.
$$ frac{d^2u}{dx^2}-xu = 0 $$
This ODE is known as Airy equation : http://mathworld.wolfram.com/AiryDifferentialEquation.html
There is no closed form for the solutions with a finite number of elementary functions.
A closed form requires special functions, namely the Airy functions Ai$(x)$ and Bi$(x)$
$$u(x)=c_1text{Ai}(x)+c_2text{Bi}(x)$$
The Airy functions are related to some other special functions, especially the Bessel functions of first kind.
To express the solutions with elementary fonctions only, infinite series are required. The calculus is rather boring. See Eqs.$(2-26)$ in the above reference.
About properties of Airy functions : http://mathworld.wolfram.com/AiryFunctions.html
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
edited Jan 7 at 10:12
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
answered Jan 7 at 9:34
JJacquelinJJacquelin
42.8k21750
42.8k21750
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
|
show 2 more comments
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
But how can I find the distributional solution? If u Is a distribution, I am not sure the canonical method is the easiest or if it even works
$endgroup$
– MargeA
Jan 7 at 9:36
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
To solve this kind of ODE, with any method you cannot avoid either an infinite series or a special function as pointed out.
$endgroup$
– JJacquelin
Jan 7 at 9:46
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
May be, your question is not well posed in the context of distributions. Elaborate the hypothesis and the equation model for the particular problem that you have to solve.
$endgroup$
– JJacquelin
Jan 7 at 9:50
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
It is an exam exercise given by my mathematics methods of physics professor. He asked to find the distributional solution of the Airy equation, nothing more
$endgroup$
– MargeA
Jan 7 at 9:52
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
$begingroup$
Then I suppose that you refer to en.wikipedia.org/wiki/… . So you have to apply your course book. In addition to your question, show what you have done (use the "edit" button) and show exactly where you are stuck.
$endgroup$
– JJacquelin
Jan 7 at 10:01
|
show 2 more comments
$begingroup$
You may try this form $u=sum_{n=-infty}^{infty}a_nx^n$, and seek for the recurrence relation between $a_n$'s.
$endgroup$
– hypernova
Jan 7 at 1:12
$begingroup$
eqworld.ipmnet.ru/en/solutions/ode/ode0202.pdf
$endgroup$
– player100
Jan 7 at 2:00
$begingroup$
Why do you need test functions? Classical solutions work just fine, and you argue like you would for any linear second-order ODE. For example, the solutions with $(u(0),u'(0))$ equal to $(1,0)$ and $(0,1)$, respectively, are linearly independent.
$endgroup$
– Hans Lundmark
Jan 7 at 7:51
$begingroup$
If you do not specify what solution method you use and exactly what the point is you wonder about, you will always get the same answer that you already got for your first identical question, before you added the Fourier calculations.
$endgroup$
– LutzL
Jan 7 at 9:51