Emden–Fowler equation how to solve?
I am trying to solve the equation
$$frac{y''}{y'}=frac{alpha}{r y^2}$$,
which is of the form of the Emden–Fowler equation.
I found a solution in chapter 2.5.2.5 of Handbook of Exact Solutions for Ordinary Differential Equations, Chapman
& Hall/CRC, however i don't understand how to derive it. I need this understanding because i get this equation from differentiating another one, and so i have to impose a constrain on the initial conditions.
The solution is apparently given by,
$$ r =pm expleft [left (inttext{d}tau tau^{-3/2}(2tau^{1/2} - 2 tau^{-1/2} +C_1)^{-1}right) +C_2right ],, tau=frac{1}{alpha y^2}$$
So my question is how to derive this result from the equation. Help much appreciated.
[UPDATE: I solved] I simply noticed that,
$$r y'' = alpha left(-frac{1}{y}right )'implies inttext{d}r ,r y'' = (ry' - y)Big|_{y_0,r_0}^{y,r} = -frac{1}{y}Big|_{y_0}^y$$
integrating once more yields to the desired result.
differential-equations nonlinear-system
asked Nov 16 at 11:20
MrFermiMr
614
|
show 1 more comment
I am trying to solve the equation
$$frac{y''}{y'}=frac{alpha}{r y^2}$$,
which is of the form of the Emden–Fowler equation.
I found a solution in chapter 2.5.2.5 of Handbook of Exact Solutions for Ordinary Differential Equations, Chapman
& Hall/CRC, however i don't understand how to derive it. I need this understanding because i get this equation from differentiating another one, and so i have to impose a constrain on the initial conditions.
The solution is apparently given by,
$$ r =pm expleft [left (inttext{d}tau tau^{-3/2}(2tau^{1/2} - 2 tau^{-1/2} +C_1)^{-1}right) +C_2right ],, tau=frac{1}{alpha y^2}$$
So my question is how to derive this result from the equation. Help much appreciated.
[UPDATE: I solved] I simply noticed that,
$$r y'' = alpha left(-frac{1}{y}right )'implies inttext{d}r ,r y'' = (ry' - y)Big|_{y_0,r_0}^{y,r} = -frac{1}{y}Big|_{y_0}^y$$
integrating once more yields to the desired result.
differential-equations nonlinear-system
asked Nov 16 at 11:20
MrFermiMr
614
Does $'$ mean $d/dr$?
– Richard Martin
Nov 16 at 11:40
@RichardMartin yes :)
– MrFermiMr
Nov 16 at 12:29
With Maple: $y left( t right) ={frac {alpha}{c_{{1}}r} left( {rm W} left({ frac {1}{alpha}{{rm e}^{{frac {r left( t+c_{{2}} right) {c_{{1}} }^{2}-alpha}{alpha}}}}}right)+1 right) } $ were $W$ is LambertW function.
– Mariusz Iwaniuk
Nov 16 at 13:00
I know what is the solution, i want to understand how it is derived.
– MrFermiMr
Nov 16 at 13:14
1
You can accept your own answer as the solution if you wish.
– Kevin
Nov 29 at 12:11
|
show 1 more comment
I am trying to solve the equation
$$frac{y''}{y'}=frac{alpha}{r y^2}$$,
which is of the form of the Emden–Fowler equation.
I found a solution in chapter 2.5.2.5 of Handbook of Exact Solutions for Ordinary Differential Equations, Chapman
& Hall/CRC, however i don't understand how to derive it. I need this understanding because i get this equation from differentiating another one, and so i have to impose a constrain on the initial conditions.
The solution is apparently given by,
$$ r =pm expleft [left (inttext{d}tau tau^{-3/2}(2tau^{1/2} - 2 tau^{-1/2} +C_1)^{-1}right) +C_2right ],, tau=frac{1}{alpha y^2}$$
So my question is how to derive this result from the equation. Help much appreciated.
[UPDATE: I solved] I simply noticed that,
$$r y'' = alpha left(-frac{1}{y}right )'implies inttext{d}r ,r y'' = (ry' - y)Big|_{y_0,r_0}^{y,r} = -frac{1}{y}Big|_{y_0}^y$$
integrating once more yields to the desired result.
differential-equations nonlinear-system
asked Nov 16 at 11:20
MrFermiMr
614
I am trying to solve the equation
$$frac{y''}{y'}=frac{alpha}{r y^2}$$,
which is of the form of the Emden–Fowler equation.
I found a solution in chapter 2.5.2.5 of Handbook of Exact Solutions for Ordinary Differential Equations, Chapman
& Hall/CRC, however i don't understand how to derive it. I need this understanding because i get this equation from differentiating another one, and so i have to impose a constrain on the initial conditions.
The solution is apparently given by,
$$ r =pm expleft [left (inttext{d}tau tau^{-3/2}(2tau^{1/2} - 2 tau^{-1/2} +C_1)^{-1}right) +C_2right ],, tau=frac{1}{alpha y^2}$$
So my question is how to derive this result from the equation. Help much appreciated.
[UPDATE: I solved] I simply noticed that,
$$r y'' = alpha left(-frac{1}{y}right )'implies inttext{d}r ,r y'' = (ry' - y)Big|_{y_0,r_0}^{y,r} = -frac{1}{y}Big|_{y_0}^y$$
integrating once more yields to the desired result.
differential-equations nonlinear-system
differential-equations nonlinear-system
asked Nov 16 at 11:20
MrFermiMr
614
asked Nov 16 at 11:20
MrFermiMr
614
edited Nov 29 at 11:10
asked Nov 16 at 11:20
MrFermiMr
614
asked Nov 16 at 11:20
MrFermiMr
614
asked Nov 16 at 11:20
MrFermiMr
614
614
Does $'$ mean $d/dr$?
– Richard Martin
Nov 16 at 11:40
@RichardMartin yes :)
– MrFermiMr
Nov 16 at 12:29
With Maple: $y left( t right) ={frac {alpha}{c_{{1}}r} left( {rm W} left({ frac {1}{alpha}{{rm e}^{{frac {r left( t+c_{{2}} right) {c_{{1}} }^{2}-alpha}{alpha}}}}}right)+1 right) } $ were $W$ is LambertW function.
– Mariusz Iwaniuk
Nov 16 at 13:00
I know what is the solution, i want to understand how it is derived.
– MrFermiMr
Nov 16 at 13:14
1
You can accept your own answer as the solution if you wish.
– Kevin
Nov 29 at 12:11
|
show 1 more comment
Does $'$ mean $d/dr$?
– Richard Martin
Nov 16 at 11:40
@RichardMartin yes :)
– MrFermiMr
Nov 16 at 12:29
With Maple: $y left( t right) ={frac {alpha}{c_{{1}}r} left( {rm W} left({ frac {1}{alpha}{{rm e}^{{frac {r left( t+c_{{2}} right) {c_{{1}} }^{2}-alpha}{alpha}}}}}right)+1 right) } $ were $W$ is LambertW function.
– Mariusz Iwaniuk
Nov 16 at 13:00
I know what is the solution, i want to understand how it is derived.
– MrFermiMr
Nov 16 at 13:14
1
You can accept your own answer as the solution if you wish.
– Kevin
Nov 29 at 12:11
Does $'$ mean $d/dr$?
– Richard Martin
Nov 16 at 11:40
Does $'$ mean $d/dr$?
– Richard Martin
Nov 16 at 11:40
@RichardMartin yes :)
– MrFermiMr
Nov 16 at 12:29
@RichardMartin yes :)
– MrFermiMr
Nov 16 at 12:29
With Maple: $y left( t right) ={frac {alpha}{c_{{1}}r} left( {rm W} left({ frac {1}{alpha}{{rm e}^{{frac {r left( t+c_{{2}} right) {c_{{1}} }^{2}-alpha}{alpha}}}}}right)+1 right) } $ were $W$ is LambertW function.
– Mariusz Iwaniuk
Nov 16 at 13:00
With Maple: $y left( t right) ={frac {alpha}{c_{{1}}r} left( {rm W} left({ frac {1}{alpha}{{rm e}^{{frac {r left( t+c_{{2}} right) {c_{{1}} }^{2}-alpha}{alpha}}}}}right)+1 right) } $ were $W$ is LambertW function.
– Mariusz Iwaniuk
Nov 16 at 13:00
I know what is the solution, i want to understand how it is derived.
– MrFermiMr
Nov 16 at 13:14
I know what is the solution, i want to understand how it is derived.
– MrFermiMr
Nov 16 at 13:14
1
1
You can accept your own answer as the solution if you wish.
– Kevin
Nov 29 at 12:11
You can accept your own answer as the solution if you wish.
– Kevin
Nov 29 at 12:11
|
show 1 more comment
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Does $'$ mean $d/dr$?
– Richard Martin
Nov 16 at 11:40
@RichardMartin yes :)
– MrFermiMr
Nov 16 at 12:29
With Maple: $y left( t right) ={frac {alpha}{c_{{1}}r} left( {rm W} left({ frac {1}{alpha}{{rm e}^{{frac {r left( t+c_{{2}} right) {c_{{1}} }^{2}-alpha}{alpha}}}}}right)+1 right) } $ were $W$ is LambertW function.
– Mariusz Iwaniuk
Nov 16 at 13:00
I know what is the solution, i want to understand how it is derived.
– MrFermiMr
Nov 16 at 13:14
1
You can accept your own answer as the solution if you wish.
– Kevin
Nov 29 at 12:11