$-y''+(1+x)y=Â lambda y$ for which value of $lambda$ the eqn has nonzero solution
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
The problem is the following
$$begincases-y''+(1+x)y=àûy\
y(0)=y(1)=0\ xin(0,1)endcases$$ For which value if $lambda$ the problem has nonzero solutions?
The answers given are
- for all $lambda<0$
- for all $lambdain[0,1]$.
- for some $lambdain(2,infty)$
- for a countable number of $lambda$'s
differential-equations pde power-series linear-pde
add a comment |Â
up vote
0
down vote
favorite
The problem is the following
$$begincases-y''+(1+x)y=àûy\
y(0)=y(1)=0\ xin(0,1)endcases$$ For which value if $lambda$ the problem has nonzero solutions?
The answers given are
- for all $lambda<0$
- for all $lambdain[0,1]$.
- for some $lambdain(2,infty)$
- for a countable number of $lambda$'s
differential-equations pde power-series linear-pde
What have you tried so far? Are there theorems in your course you can use, which might help you? For example something about eigenvalues of linear elliptic operators?
â humanStampedist
Aug 22 at 8:25
What about your work?
â Cesareo
Aug 22 at 8:31
Same question in math.stackexchange.com/questions/1838947/â¦, math.stackexchange.com/questions/2318378/â¦, similar question in math.stackexchange.com/questions/2686316/â¦.
â LutzL
Aug 22 at 9:17
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The problem is the following
$$begincases-y''+(1+x)y=àûy\
y(0)=y(1)=0\ xin(0,1)endcases$$ For which value if $lambda$ the problem has nonzero solutions?
The answers given are
- for all $lambda<0$
- for all $lambdain[0,1]$.
- for some $lambdain(2,infty)$
- for a countable number of $lambda$'s
differential-equations pde power-series linear-pde
The problem is the following
$$begincases-y''+(1+x)y=àûy\
y(0)=y(1)=0\ xin(0,1)endcases$$ For which value if $lambda$ the problem has nonzero solutions?
The answers given are
- for all $lambda<0$
- for all $lambdain[0,1]$.
- for some $lambdain(2,infty)$
- for a countable number of $lambda$'s
differential-equations pde power-series linear-pde
edited Aug 22 at 8:42
Davide Morgante
2,480423
2,480423
asked Aug 22 at 8:01
user532616
12
12
What have you tried so far? Are there theorems in your course you can use, which might help you? For example something about eigenvalues of linear elliptic operators?
â humanStampedist
Aug 22 at 8:25
What about your work?
â Cesareo
Aug 22 at 8:31
Same question in math.stackexchange.com/questions/1838947/â¦, math.stackexchange.com/questions/2318378/â¦, similar question in math.stackexchange.com/questions/2686316/â¦.
â LutzL
Aug 22 at 9:17
add a comment |Â
What have you tried so far? Are there theorems in your course you can use, which might help you? For example something about eigenvalues of linear elliptic operators?
â humanStampedist
Aug 22 at 8:25
What about your work?
â Cesareo
Aug 22 at 8:31
Same question in math.stackexchange.com/questions/1838947/â¦, math.stackexchange.com/questions/2318378/â¦, similar question in math.stackexchange.com/questions/2686316/â¦.
â LutzL
Aug 22 at 9:17
What have you tried so far? Are there theorems in your course you can use, which might help you? For example something about eigenvalues of linear elliptic operators?
â humanStampedist
Aug 22 at 8:25
What have you tried so far? Are there theorems in your course you can use, which might help you? For example something about eigenvalues of linear elliptic operators?
â humanStampedist
Aug 22 at 8:25
What about your work?
â Cesareo
Aug 22 at 8:31
What about your work?
â Cesareo
Aug 22 at 8:31
Same question in math.stackexchange.com/questions/1838947/â¦, math.stackexchange.com/questions/2318378/â¦, similar question in math.stackexchange.com/questions/2686316/â¦.
â LutzL
Aug 22 at 9:17
Same question in math.stackexchange.com/questions/1838947/â¦, math.stackexchange.com/questions/2318378/â¦, similar question in math.stackexchange.com/questions/2686316/â¦.
â LutzL
Aug 22 at 9:17
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
Just a partial answer: Multiply your equation with $y$ and integrate. Then you obtain
$$-int_0^1 y'' ydx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx$$
Now use integration by parts on the first integral to get
$$-[y'y]_0^1 + int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx,$$
which simplifies by the boundary values to
$$int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx.$$
$xin[0,1]$ further yields
$$int_0^1y^2, dxleq lambda int_0^1 y^2dx,$$
which exclude option 1) and 2). For $lambda=1$ you need go through the argument again and proceed by contradiction.
Existence can be done in a more general context. Look up Gilbarg/Trudinger's book on elliptic partial differential equations of second order. It should be Theorem 8.37.
This is a reference to the book:
https://www.springer.com/de/book/9783540411604
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Just a partial answer: Multiply your equation with $y$ and integrate. Then you obtain
$$-int_0^1 y'' ydx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx$$
Now use integration by parts on the first integral to get
$$-[y'y]_0^1 + int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx,$$
which simplifies by the boundary values to
$$int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx.$$
$xin[0,1]$ further yields
$$int_0^1y^2, dxleq lambda int_0^1 y^2dx,$$
which exclude option 1) and 2). For $lambda=1$ you need go through the argument again and proceed by contradiction.
Existence can be done in a more general context. Look up Gilbarg/Trudinger's book on elliptic partial differential equations of second order. It should be Theorem 8.37.
This is a reference to the book:
https://www.springer.com/de/book/9783540411604
add a comment |Â
up vote
1
down vote
Just a partial answer: Multiply your equation with $y$ and integrate. Then you obtain
$$-int_0^1 y'' ydx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx$$
Now use integration by parts on the first integral to get
$$-[y'y]_0^1 + int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx,$$
which simplifies by the boundary values to
$$int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx.$$
$xin[0,1]$ further yields
$$int_0^1y^2, dxleq lambda int_0^1 y^2dx,$$
which exclude option 1) and 2). For $lambda=1$ you need go through the argument again and proceed by contradiction.
Existence can be done in a more general context. Look up Gilbarg/Trudinger's book on elliptic partial differential equations of second order. It should be Theorem 8.37.
This is a reference to the book:
https://www.springer.com/de/book/9783540411604
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Just a partial answer: Multiply your equation with $y$ and integrate. Then you obtain
$$-int_0^1 y'' ydx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx$$
Now use integration by parts on the first integral to get
$$-[y'y]_0^1 + int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx,$$
which simplifies by the boundary values to
$$int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx.$$
$xin[0,1]$ further yields
$$int_0^1y^2, dxleq lambda int_0^1 y^2dx,$$
which exclude option 1) and 2). For $lambda=1$ you need go through the argument again and proceed by contradiction.
Existence can be done in a more general context. Look up Gilbarg/Trudinger's book on elliptic partial differential equations of second order. It should be Theorem 8.37.
This is a reference to the book:
https://www.springer.com/de/book/9783540411604
Just a partial answer: Multiply your equation with $y$ and integrate. Then you obtain
$$-int_0^1 y'' ydx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx$$
Now use integration by parts on the first integral to get
$$-[y'y]_0^1 + int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx,$$
which simplifies by the boundary values to
$$int_0^1(y')^2dx + int_0^1(1+x)y^2dx = lambda int_0^1y^2dx.$$
$xin[0,1]$ further yields
$$int_0^1y^2, dxleq lambda int_0^1 y^2dx,$$
which exclude option 1) and 2). For $lambda=1$ you need go through the argument again and proceed by contradiction.
Existence can be done in a more general context. Look up Gilbarg/Trudinger's book on elliptic partial differential equations of second order. It should be Theorem 8.37.
This is a reference to the book:
https://www.springer.com/de/book/9783540411604
answered Aug 22 at 8:45
humanStampedist
2,036213
2,036213
add a comment |Â
add a comment |Â
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%2f2890758%2fy1xy-lambda-y-for-which-value-of-lambda-the-eqn-has-nonzero-soluti%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
What have you tried so far? Are there theorems in your course you can use, which might help you? For example something about eigenvalues of linear elliptic operators?
â humanStampedist
Aug 22 at 8:25
What about your work?
â Cesareo
Aug 22 at 8:31
Same question in math.stackexchange.com/questions/1838947/â¦, math.stackexchange.com/questions/2318378/â¦, similar question in math.stackexchange.com/questions/2686316/â¦.
â LutzL
Aug 22 at 9:17