Mixing
Increase of free parameters in perturbed solution of an ODE
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Lets take the ODE
beginequation
ODE0: qquad fracpartial y_0partial x = F(y_0,x)
endequation
with known solution $y_0(c_0)$, where $c_0$ is (are) the free parameter(s) that I will fix by exploiting the initial conditions.
Lets now take
beginequation labeleq:q_1
ODE1: qquad fracpartial ypartial x = F(y,x) + beta ; G(y,x) ; ,
endequation
which instead I am not able to solve exactly. It looks reasonable to me to take
$y simeq y_0(c_0) + beta , y_1 ; ,$
as an approximated solution of this new ODE1.
I now suppose to be able to rewrite ODE1 as
$fracpartial y_1partial x simeq H(y_1,y_0,x) ; .$
I also suppose I am able to find a solution $y_1(c_1)$ for this one.
Conclusively I have
$y(c_0,c_1) simeq y_0(c_0) + beta , y_1(c_1) ; ,$
which is an approximated solution for ODE1. However $y$ depends on an higher number of free parameters w.r.t. $y_0$ ($c_0$ and $c_1$, instead of just $c_0$), despite being a solution of an equation of the same order (i.e. ODE0).
This seems to not add up from any side I look at it.
Shouldn't the amount of free parameters remain invariant, and equal to the order of the ODE?
Is there some fallacy in my reasoning?
Thank you very much in advance for all the help you may give.
NOTE:
I tried defining the problem in a way as general as possible, so to make it interesting for a wide group of people.
The problem is indeed general in my mind, but it arose in a pretty specific situation.
differential-equations initial-value-problems non-linear-dynamics
asked Jul 22 at 18:50
CosimoDS
64
add a comment |Â
up vote
0
down vote
favorite
Lets take the ODE
beginequation
ODE0: qquad fracpartial y_0partial x = F(y_0,x)
endequation
with known solution $y_0(c_0)$, where $c_0$ is (are) the free parameter(s) that I will fix by exploiting the initial conditions.
Lets now take
beginequation labeleq:q_1
ODE1: qquad fracpartial ypartial x = F(y,x) + beta ; G(y,x) ; ,
endequation
which instead I am not able to solve exactly. It looks reasonable to me to take
$y simeq y_0(c_0) + beta , y_1 ; ,$
as an approximated solution of this new ODE1.
I now suppose to be able to rewrite ODE1 as
$fracpartial y_1partial x simeq H(y_1,y_0,x) ; .$
I also suppose I am able to find a solution $y_1(c_1)$ for this one.
Conclusively I have
$y(c_0,c_1) simeq y_0(c_0) + beta , y_1(c_1) ; ,$
which is an approximated solution for ODE1. However $y$ depends on an higher number of free parameters w.r.t. $y_0$ ($c_0$ and $c_1$, instead of just $c_0$), despite being a solution of an equation of the same order (i.e. ODE0).
This seems to not add up from any side I look at it.
Shouldn't the amount of free parameters remain invariant, and equal to the order of the ODE?
Is there some fallacy in my reasoning?
Thank you very much in advance for all the help you may give.
NOTE:
I tried defining the problem in a way as general as possible, so to make it interesting for a wide group of people.
The problem is indeed general in my mind, but it arose in a pretty specific situation.
differential-equations initial-value-problems non-linear-dynamics
asked Jul 22 at 18:50
CosimoDS
64
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Lets take the ODE
beginequation
ODE0: qquad fracpartial y_0partial x = F(y_0,x)
endequation
with known solution $y_0(c_0)$, where $c_0$ is (are) the free parameter(s) that I will fix by exploiting the initial conditions.
Lets now take
beginequation labeleq:q_1
ODE1: qquad fracpartial ypartial x = F(y,x) + beta ; G(y,x) ; ,
endequation
which instead I am not able to solve exactly. It looks reasonable to me to take
$y simeq y_0(c_0) + beta , y_1 ; ,$
as an approximated solution of this new ODE1.
I now suppose to be able to rewrite ODE1 as
$fracpartial y_1partial x simeq H(y_1,y_0,x) ; .$
I also suppose I am able to find a solution $y_1(c_1)$ for this one.
Conclusively I have
$y(c_0,c_1) simeq y_0(c_0) + beta , y_1(c_1) ; ,$
which is an approximated solution for ODE1. However $y$ depends on an higher number of free parameters w.r.t. $y_0$ ($c_0$ and $c_1$, instead of just $c_0$), despite being a solution of an equation of the same order (i.e. ODE0).
This seems to not add up from any side I look at it.
Shouldn't the amount of free parameters remain invariant, and equal to the order of the ODE?
Is there some fallacy in my reasoning?
Thank you very much in advance for all the help you may give.
NOTE:
I tried defining the problem in a way as general as possible, so to make it interesting for a wide group of people.
The problem is indeed general in my mind, but it arose in a pretty specific situation.
differential-equations initial-value-problems non-linear-dynamics
asked Jul 22 at 18:50
CosimoDS
64
Lets take the ODE
beginequation
ODE0: qquad fracpartial y_0partial x = F(y_0,x)
endequation
with known solution $y_0(c_0)$, where $c_0$ is (are) the free parameter(s) that I will fix by exploiting the initial conditions.
Lets now take
beginequation labeleq:q_1
ODE1: qquad fracpartial ypartial x = F(y,x) + beta ; G(y,x) ; ,
endequation
which instead I am not able to solve exactly. It looks reasonable to me to take
$y simeq y_0(c_0) + beta , y_1 ; ,$
as an approximated solution of this new ODE1.
I now suppose to be able to rewrite ODE1 as
$fracpartial y_1partial x simeq H(y_1,y_0,x) ; .$
I also suppose I am able to find a solution $y_1(c_1)$ for this one.
Conclusively I have
$y(c_0,c_1) simeq y_0(c_0) + beta , y_1(c_1) ; ,$
which is an approximated solution for ODE1. However $y$ depends on an higher number of free parameters w.r.t. $y_0$ ($c_0$ and $c_1$, instead of just $c_0$), despite being a solution of an equation of the same order (i.e. ODE0).
This seems to not add up from any side I look at it.
Shouldn't the amount of free parameters remain invariant, and equal to the order of the ODE?
Is there some fallacy in my reasoning?
Thank you very much in advance for all the help you may give.
NOTE:
I tried defining the problem in a way as general as possible, so to make it interesting for a wide group of people.
The problem is indeed general in my mind, but it arose in a pretty specific situation.
differential-equations initial-value-problems non-linear-dynamics
asked Jul 22 at 18:50
CosimoDS
64
asked Jul 22 at 18:50
CosimoDS
64
asked Jul 22 at 18:50
CosimoDS
64
asked Jul 22 at 18:50
CosimoDS
64
64
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f2859668%2fincrease-of-free-parameters-in-perturbed-solution-of-an-ode%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