Solution of the second-order ODE with polynomial coefficients $(a-x),x, y''(x)+left(fraca2-2 ,xright)y'(x)+lambda ,x^2y(x)=0$.
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
1
I am searching for solutions of the following second order differential equation:
$$(a-x),x, y''(x)+left(fraca2-2 ,xright)y'(x)+lambda ,x^2y(x)=0,qquad (0<x<a)$$
with $y(a)=0$ and $y'(0)=0$.
As far as I know, there is no finite polynomial solution. In a book I have found a solution when the factor $x^2$ in the last term is replaced by 1.
EDIT: Boundedness of $y(x)$ for $xto 0$ is an alternative condition (instead of $y'(0)=0$).
differential-equations
edited Aug 28 at 16:26
Nosrati
21.9k51747
asked Aug 27 at 17:04
kaffeeauf
1409
 |Â
show 2 more comments
up vote
2
down vote
favorite
1
I am searching for solutions of the following second order differential equation:
$$(a-x),x, y''(x)+left(fraca2-2 ,xright)y'(x)+lambda ,x^2y(x)=0,qquad (0<x<a)$$
with $y(a)=0$ and $y'(0)=0$.
As far as I know, there is no finite polynomial solution. In a book I have found a solution when the factor $x^2$ in the last term is replaced by 1.
EDIT: Boundedness of $y(x)$ for $xto 0$ is an alternative condition (instead of $y'(0)=0$).
differential-equations
edited Aug 28 at 16:26
Nosrati
21.9k51747
asked Aug 27 at 17:04
kaffeeauf
1409
2
If the Wolfram Dev Platform doesn't give you a closed-form solution, there probably isn't one. If you have to have an exact solution, then a series method is your only option, most likely. Otherwise, of course, you could always go numerical.
– Adrian Keister
Aug 27 at 17:12
1
Even with series, the condition $y(a)=0$ would make the problem very difficult.
– Claude Leibovici
Aug 28 at 4:04
@Claude is there an alternative condition which brings out a closed form solution?
– kaffeeauf
Aug 28 at 5:27
1
Maple 2018.1 gives general solution with HeunC function adding boundary conditions: $y(a)=0$,$y'(0)=0$ spit up one solution: $y(x)=0$
– Mariusz Iwaniuk
Aug 28 at 10:22
1
$$y left( x right) =frac it _C1,rm e^sqrt lambdax sqrt x-ait HeunC left( 2,asqrt lambda,-1/2,-1/2,-lambda ,a^2,1/8,frac xa right) +frac it _C2,rm e^ sqrt lambdaxsqrt xsqrt x-ait HeunC left( 2,asqrt lambda,1/2,-1/2,-lambda,a^2,1/8,frac xa right) $$ where: $_C1$,$_C2$ integration constans.
– Mariusz Iwaniuk
Aug 28 at 16:38
 |Â
show 2 more comments
up vote
2
down vote
favorite
1
up vote
2
down vote
favorite
1
1
I am searching for solutions of the following second order differential equation:
$$(a-x),x, y''(x)+left(fraca2-2 ,xright)y'(x)+lambda ,x^2y(x)=0,qquad (0<x<a)$$
with $y(a)=0$ and $y'(0)=0$.
As far as I know, there is no finite polynomial solution. In a book I have found a solution when the factor $x^2$ in the last term is replaced by 1.
EDIT: Boundedness of $y(x)$ for $xto 0$ is an alternative condition (instead of $y'(0)=0$).
differential-equations
edited Aug 28 at 16:26
Nosrati
21.9k51747
asked Aug 27 at 17:04
kaffeeauf
1409
I am searching for solutions of the following second order differential equation:
$$(a-x),x, y''(x)+left(fraca2-2 ,xright)y'(x)+lambda ,x^2y(x)=0,qquad (0<x<a)$$
with $y(a)=0$ and $y'(0)=0$.
As far as I know, there is no finite polynomial solution. In a book I have found a solution when the factor $x^2$ in the last term is replaced by 1.
EDIT: Boundedness of $y(x)$ for $xto 0$ is an alternative condition (instead of $y'(0)=0$).
differential-equations
edited Aug 28 at 16:26
Nosrati
21.9k51747
asked Aug 27 at 17:04
kaffeeauf
1409
edited Aug 28 at 16:26
Nosrati
21.9k51747
edited Aug 28 at 16:26
Nosrati
21.9k51747
edited Aug 28 at 16:26
Nosrati
21.9k51747
21.9k51747
asked Aug 27 at 17:04
kaffeeauf
1409
asked Aug 27 at 17:04
kaffeeauf
1409
asked Aug 27 at 17:04
kaffeeauf
1409
1409
2
If the Wolfram Dev Platform doesn't give you a closed-form solution, there probably isn't one. If you have to have an exact solution, then a series method is your only option, most likely. Otherwise, of course, you could always go numerical.
– Adrian Keister
Aug 27 at 17:12
1
Even with series, the condition $y(a)=0$ would make the problem very difficult.
– Claude Leibovici
Aug 28 at 4:04
@Claude is there an alternative condition which brings out a closed form solution?
– kaffeeauf
Aug 28 at 5:27
1
Maple 2018.1 gives general solution with HeunC function adding boundary conditions: $y(a)=0$,$y'(0)=0$ spit up one solution: $y(x)=0$
– Mariusz Iwaniuk
Aug 28 at 10:22
1
$$y left( x right) =frac it _C1,rm e^sqrt lambdax sqrt x-ait HeunC left( 2,asqrt lambda,-1/2,-1/2,-lambda ,a^2,1/8,frac xa right) +frac it _C2,rm e^ sqrt lambdaxsqrt xsqrt x-ait HeunC left( 2,asqrt lambda,1/2,-1/2,-lambda,a^2,1/8,frac xa right) $$ where: $_C1$,$_C2$ integration constans.
– Mariusz Iwaniuk
Aug 28 at 16:38
 |Â
show 2 more comments
2
If the Wolfram Dev Platform doesn't give you a closed-form solution, there probably isn't one. If you have to have an exact solution, then a series method is your only option, most likely. Otherwise, of course, you could always go numerical.
– Adrian Keister
Aug 27 at 17:12
1
Even with series, the condition $y(a)=0$ would make the problem very difficult.
– Claude Leibovici
Aug 28 at 4:04
@Claude is there an alternative condition which brings out a closed form solution?
– kaffeeauf
Aug 28 at 5:27
1
Maple 2018.1 gives general solution with HeunC function adding boundary conditions: $y(a)=0$,$y'(0)=0$ spit up one solution: $y(x)=0$
– Mariusz Iwaniuk
Aug 28 at 10:22
1
$$y left( x right) =frac it _C1,rm e^sqrt lambdax sqrt x-ait HeunC left( 2,asqrt lambda,-1/2,-1/2,-lambda ,a^2,1/8,frac xa right) +frac it _C2,rm e^ sqrt lambdaxsqrt xsqrt x-ait HeunC left( 2,asqrt lambda,1/2,-1/2,-lambda,a^2,1/8,frac xa right) $$ where: $_C1$,$_C2$ integration constans.
– Mariusz Iwaniuk
Aug 28 at 16:38
2
2
If the Wolfram Dev Platform doesn't give you a closed-form solution, there probably isn't one. If you have to have an exact solution, then a series method is your only option, most likely. Otherwise, of course, you could always go numerical.
– Adrian Keister
Aug 27 at 17:12
If the Wolfram Dev Platform doesn't give you a closed-form solution, there probably isn't one. If you have to have an exact solution, then a series method is your only option, most likely. Otherwise, of course, you could always go numerical.
– Adrian Keister
Aug 27 at 17:12
1
1
Even with series, the condition $y(a)=0$ would make the problem very difficult.
– Claude Leibovici
Aug 28 at 4:04
Even with series, the condition $y(a)=0$ would make the problem very difficult.
– Claude Leibovici
Aug 28 at 4:04
@Claude is there an alternative condition which brings out a closed form solution?
– kaffeeauf
Aug 28 at 5:27
@Claude is there an alternative condition which brings out a closed form solution?
– kaffeeauf
Aug 28 at 5:27
1
1
Maple 2018.1 gives general solution with HeunC function adding boundary conditions: $y(a)=0$,$y'(0)=0$ spit up one solution: $y(x)=0$
– Mariusz Iwaniuk
Aug 28 at 10:22
Maple 2018.1 gives general solution with HeunC function adding boundary conditions: $y(a)=0$,$y'(0)=0$ spit up one solution: $y(x)=0$
– Mariusz Iwaniuk
Aug 28 at 10:22
1
1
$$y left( x right) =frac it _C1,rm e^sqrt lambdax sqrt x-ait HeunC left( 2,asqrt lambda,-1/2,-1/2,-lambda ,a^2,1/8,frac xa right) +frac it _C2,rm e^ sqrt lambdaxsqrt xsqrt x-ait HeunC left( 2,asqrt lambda,1/2,-1/2,-lambda,a^2,1/8,frac xa right) $$ where: $_C1$,$_C2$ integration constans.
– Mariusz Iwaniuk
Aug 28 at 16:38
$$y left( x right) =frac it _C1,rm e^sqrt lambdax sqrt x-ait HeunC left( 2,asqrt lambda,-1/2,-1/2,-lambda ,a^2,1/8,frac xa right) +frac it _C2,rm e^ sqrt lambdaxsqrt xsqrt x-ait HeunC left( 2,asqrt lambda,1/2,-1/2,-lambda,a^2,1/8,frac xa right) $$ where: $_C1$,$_C2$ integration constans.
– Mariusz Iwaniuk
Aug 28 at 16:38
 |Â
show 2 more comments
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Â
draft saved
draft discarded
Â
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2896411%2fsolution-of-the-second-order-ode-with-polynomial-coefficients-a-x-x-yx%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
2
If the Wolfram Dev Platform doesn't give you a closed-form solution, there probably isn't one. If you have to have an exact solution, then a series method is your only option, most likely. Otherwise, of course, you could always go numerical.
– Adrian Keister
Aug 27 at 17:12
1
Even with series, the condition $y(a)=0$ would make the problem very difficult.
– Claude Leibovici
Aug 28 at 4:04
@Claude is there an alternative condition which brings out a closed form solution?
– kaffeeauf
Aug 28 at 5:27
1
Maple 2018.1 gives general solution with HeunC function adding boundary conditions: $y(a)=0$,$y'(0)=0$ spit up one solution: $y(x)=0$
– Mariusz Iwaniuk
Aug 28 at 10:22
1
$$y left( x right) =frac it _C1,rm e^sqrt lambdax sqrt x-ait HeunC left( 2,asqrt lambda,-1/2,-1/2,-lambda ,a^2,1/8,frac xa right) +frac it _C2,rm e^ sqrt lambdaxsqrt xsqrt x-ait HeunC left( 2,asqrt lambda,1/2,-1/2,-lambda,a^2,1/8,frac xa right) $$ where: $_C1$,$_C2$ integration constans.
– Mariusz Iwaniuk
Aug 28 at 16:38