Mixing
What is the solution to $y*y’’=sin(x)$? [closed]
$begingroup$
I saw this differential equation in a khan academy video. I’m still in high school and I have just began to learn a lot about differential equations, so i wanted to see if i could solve this. I don’t really know how to though. Apologies if it’s a stupid question, that’s just me.
The differential equation is $$yfrac{d^2y}{dx^2}=sin(x)$$. If anybody can tell me the answer to this, and how to get it, if it is solvable, i’d love to know.
Link to the video, the uploader mentions it just at the end but doesn’t really say much about it: https://youtu.be/-_POEWfygmU
ordinary-differential-equations
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
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closed as off-topic by Saad, Claude Leibovici, Leucippus, Abcd, Cesareo Jan 7 at 7:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Claude Leibovici, Leucippus, Abcd, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 1 more comment
$begingroup$
I saw this differential equation in a khan academy video. I’m still in high school and I have just began to learn a lot about differential equations, so i wanted to see if i could solve this. I don’t really know how to though. Apologies if it’s a stupid question, that’s just me.
The differential equation is $$yfrac{d^2y}{dx^2}=sin(x)$$. If anybody can tell me the answer to this, and how to get it, if it is solvable, i’d love to know.
Link to the video, the uploader mentions it just at the end but doesn’t really say much about it: https://youtu.be/-_POEWfygmU
ordinary-differential-equations
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
$endgroup$
closed as off-topic by Saad, Claude Leibovici, Leucippus, Abcd, Cesareo Jan 7 at 7:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Claude Leibovici, Leucippus, Abcd, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
Add a link for this video. I am not sure that this equation has an explicit solution.
$endgroup$
– Claude Leibovici
Jan 7 at 4:02
$begingroup$
@Claude Leibovici youtu.be/-_POEWfygmU He mentions it just at the end of the video. I’m guessing there isn’t an explicit solution because it’s non-linear. It’s deceptively simple looking so i though i’d give it a try lol.
$endgroup$
– Thatpotatoisaspy
Jan 7 at 4:16
2
$begingroup$
In general, nonlinear differential equations (like the one you are asking about) are hard to solve. The video seems to discuss only linear differential equations.
$endgroup$
– D.B.
Jan 7 at 4:22
1
$begingroup$
Even the time-homogeneous $ycdot y'' = -1$ looks nasty.
$endgroup$
– Henning Makholm
Jan 7 at 4:29
1
$begingroup$
@HenningMakholm I think with that one, you could do $$y'cdot y'' = -frac{y'}{y}$$and then integrate. That at least looks a bit solvable.
$endgroup$
– John Doe
Jan 7 at 5:05
|
show 1 more comment
$begingroup$
I saw this differential equation in a khan academy video. I’m still in high school and I have just began to learn a lot about differential equations, so i wanted to see if i could solve this. I don’t really know how to though. Apologies if it’s a stupid question, that’s just me.
The differential equation is $$yfrac{d^2y}{dx^2}=sin(x)$$. If anybody can tell me the answer to this, and how to get it, if it is solvable, i’d love to know.
Link to the video, the uploader mentions it just at the end but doesn’t really say much about it: https://youtu.be/-_POEWfygmU
ordinary-differential-equations
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
$endgroup$
I saw this differential equation in a khan academy video. I’m still in high school and I have just began to learn a lot about differential equations, so i wanted to see if i could solve this. I don’t really know how to though. Apologies if it’s a stupid question, that’s just me.
The differential equation is $$yfrac{d^2y}{dx^2}=sin(x)$$. If anybody can tell me the answer to this, and how to get it, if it is solvable, i’d love to know.
Link to the video, the uploader mentions it just at the end but doesn’t really say much about it: https://youtu.be/-_POEWfygmU
ordinary-differential-equations
ordinary-differential-equations
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
edited Jan 7 at 4:20
Thatpotatoisaspy
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
asked Jan 7 at 3:43
ThatpotatoisaspyThatpotatoisaspy
1144
1144
closed as off-topic by Saad, Claude Leibovici, Leucippus, Abcd, Cesareo Jan 7 at 7:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Claude Leibovici, Leucippus, Abcd, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, Claude Leibovici, Leucippus, Abcd, Cesareo Jan 7 at 7:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Claude Leibovici, Leucippus, Abcd, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
Add a link for this video. I am not sure that this equation has an explicit solution.
$endgroup$
– Claude Leibovici
Jan 7 at 4:02
$begingroup$
@Claude Leibovici youtu.be/-_POEWfygmU He mentions it just at the end of the video. I’m guessing there isn’t an explicit solution because it’s non-linear. It’s deceptively simple looking so i though i’d give it a try lol.
$endgroup$
– Thatpotatoisaspy
Jan 7 at 4:16
2
$begingroup$
In general, nonlinear differential equations (like the one you are asking about) are hard to solve. The video seems to discuss only linear differential equations.
$endgroup$
– D.B.
Jan 7 at 4:22
1
$begingroup$
Even the time-homogeneous $ycdot y'' = -1$ looks nasty.
$endgroup$
– Henning Makholm
Jan 7 at 4:29
1
$begingroup$
@HenningMakholm I think with that one, you could do $$y'cdot y'' = -frac{y'}{y}$$and then integrate. That at least looks a bit solvable.
$endgroup$
– John Doe
Jan 7 at 5:05
|
show 1 more comment
1
$begingroup$
Add a link for this video. I am not sure that this equation has an explicit solution.
$endgroup$
– Claude Leibovici
Jan 7 at 4:02
$begingroup$
@Claude Leibovici youtu.be/-_POEWfygmU He mentions it just at the end of the video. I’m guessing there isn’t an explicit solution because it’s non-linear. It’s deceptively simple looking so i though i’d give it a try lol.
$endgroup$
– Thatpotatoisaspy
Jan 7 at 4:16
2
$begingroup$
In general, nonlinear differential equations (like the one you are asking about) are hard to solve. The video seems to discuss only linear differential equations.
$endgroup$
– D.B.
Jan 7 at 4:22
1
$begingroup$
Even the time-homogeneous $ycdot y'' = -1$ looks nasty.
$endgroup$
– Henning Makholm
Jan 7 at 4:29
1
$begingroup$
@HenningMakholm I think with that one, you could do $$y'cdot y'' = -frac{y'}{y}$$and then integrate. That at least looks a bit solvable.
$endgroup$
– John Doe
Jan 7 at 5:05
1
1
$begingroup$
Add a link for this video. I am not sure that this equation has an explicit solution.
$endgroup$
– Claude Leibovici
Jan 7 at 4:02
$begingroup$
Add a link for this video. I am not sure that this equation has an explicit solution.
$endgroup$
– Claude Leibovici
Jan 7 at 4:02
$begingroup$
@Claude Leibovici youtu.be/-_POEWfygmU He mentions it just at the end of the video. I’m guessing there isn’t an explicit solution because it’s non-linear. It’s deceptively simple looking so i though i’d give it a try lol.
$endgroup$
– Thatpotatoisaspy
Jan 7 at 4:16
$begingroup$
@Claude Leibovici youtu.be/-_POEWfygmU He mentions it just at the end of the video. I’m guessing there isn’t an explicit solution because it’s non-linear. It’s deceptively simple looking so i though i’d give it a try lol.
$endgroup$
– Thatpotatoisaspy
Jan 7 at 4:16
2
2
$begingroup$
In general, nonlinear differential equations (like the one you are asking about) are hard to solve. The video seems to discuss only linear differential equations.
$endgroup$
– D.B.
Jan 7 at 4:22
$begingroup$
In general, nonlinear differential equations (like the one you are asking about) are hard to solve. The video seems to discuss only linear differential equations.
$endgroup$
– D.B.
Jan 7 at 4:22
1
1
$begingroup$
Even the time-homogeneous $ycdot y'' = -1$ looks nasty.
$endgroup$
– Henning Makholm
Jan 7 at 4:29
$begingroup$
Even the time-homogeneous $ycdot y'' = -1$ looks nasty.
$endgroup$
– Henning Makholm
Jan 7 at 4:29
1
1
$begingroup$
@HenningMakholm I think with that one, you could do $$y'cdot y'' = -frac{y'}{y}$$and then integrate. That at least looks a bit solvable.
$endgroup$
– John Doe
Jan 7 at 5:05
$begingroup$
@HenningMakholm I think with that one, you could do $$y'cdot y'' = -frac{y'}{y}$$and then integrate. That at least looks a bit solvable.
$endgroup$
– John Doe
Jan 7 at 5:05
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
While an analytic expression is not feasible for general nonlinear differential equations, one can still apply many of the standard techniques used to solve linear differential equations to obtain approximate expressions. E.g., it's usually quite easy to obtain a solution in the form of a series expansions. In this case, we can try a series expansion around $x = 0$, but we have to note that $sin(x)$ is an odd function while the product $y y''$ cannot be odd if it is analytic in a neighborhood of $x = 0$.
We can therefore try to find a solution of the form $y(x) = x^{alpha} y_{text{an}}(x)$, where $ y_{text{an}}(x)$ is analytic. This then leads to a series expansion of the form:
$$y(x) = frac{2}{sqrt{3}}x^{frac{3}{2}}left[1 + a_2 x^2 + a_4 x^4 +cdotsright]$$
The coefficients are then easily evaluated by inserting this in the differential equation and equating equal powers of $x$, the first few series coefficients are given by:
$$
begin{split}
a_2 &= -frac{1}{76}\
a_4 &= frac{547}{2945760}\
a_6 &=-frac{19079}{14104298880}
end{split}
$$
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
While an analytic expression is not feasible for general nonlinear differential equations, one can still apply many of the standard techniques used to solve linear differential equations to obtain approximate expressions. E.g., it's usually quite easy to obtain a solution in the form of a series expansions. In this case, we can try a series expansion around $x = 0$, but we have to note that $sin(x)$ is an odd function while the product $y y''$ cannot be odd if it is analytic in a neighborhood of $x = 0$.
We can therefore try to find a solution of the form $y(x) = x^{alpha} y_{text{an}}(x)$, where $ y_{text{an}}(x)$ is analytic. This then leads to a series expansion of the form:
$$y(x) = frac{2}{sqrt{3}}x^{frac{3}{2}}left[1 + a_2 x^2 + a_4 x^4 +cdotsright]$$
The coefficients are then easily evaluated by inserting this in the differential equation and equating equal powers of $x$, the first few series coefficients are given by:
$$
begin{split}
a_2 &= -frac{1}{76}\
a_4 &= frac{547}{2945760}\
a_6 &=-frac{19079}{14104298880}
end{split}
$$
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
$endgroup$
add a comment |
$begingroup$
While an analytic expression is not feasible for general nonlinear differential equations, one can still apply many of the standard techniques used to solve linear differential equations to obtain approximate expressions. E.g., it's usually quite easy to obtain a solution in the form of a series expansions. In this case, we can try a series expansion around $x = 0$, but we have to note that $sin(x)$ is an odd function while the product $y y''$ cannot be odd if it is analytic in a neighborhood of $x = 0$.
We can therefore try to find a solution of the form $y(x) = x^{alpha} y_{text{an}}(x)$, where $ y_{text{an}}(x)$ is analytic. This then leads to a series expansion of the form:
$$y(x) = frac{2}{sqrt{3}}x^{frac{3}{2}}left[1 + a_2 x^2 + a_4 x^4 +cdotsright]$$
The coefficients are then easily evaluated by inserting this in the differential equation and equating equal powers of $x$, the first few series coefficients are given by:
$$
begin{split}
a_2 &= -frac{1}{76}\
a_4 &= frac{547}{2945760}\
a_6 &=-frac{19079}{14104298880}
end{split}
$$
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
$endgroup$
add a comment |
$begingroup$
While an analytic expression is not feasible for general nonlinear differential equations, one can still apply many of the standard techniques used to solve linear differential equations to obtain approximate expressions. E.g., it's usually quite easy to obtain a solution in the form of a series expansions. In this case, we can try a series expansion around $x = 0$, but we have to note that $sin(x)$ is an odd function while the product $y y''$ cannot be odd if it is analytic in a neighborhood of $x = 0$.
We can therefore try to find a solution of the form $y(x) = x^{alpha} y_{text{an}}(x)$, where $ y_{text{an}}(x)$ is analytic. This then leads to a series expansion of the form:
$$y(x) = frac{2}{sqrt{3}}x^{frac{3}{2}}left[1 + a_2 x^2 + a_4 x^4 +cdotsright]$$
The coefficients are then easily evaluated by inserting this in the differential equation and equating equal powers of $x$, the first few series coefficients are given by:
$$
begin{split}
a_2 &= -frac{1}{76}\
a_4 &= frac{547}{2945760}\
a_6 &=-frac{19079}{14104298880}
end{split}
$$
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
$endgroup$
While an analytic expression is not feasible for general nonlinear differential equations, one can still apply many of the standard techniques used to solve linear differential equations to obtain approximate expressions. E.g., it's usually quite easy to obtain a solution in the form of a series expansions. In this case, we can try a series expansion around $x = 0$, but we have to note that $sin(x)$ is an odd function while the product $y y''$ cannot be odd if it is analytic in a neighborhood of $x = 0$.
We can therefore try to find a solution of the form $y(x) = x^{alpha} y_{text{an}}(x)$, where $ y_{text{an}}(x)$ is analytic. This then leads to a series expansion of the form:
$$y(x) = frac{2}{sqrt{3}}x^{frac{3}{2}}left[1 + a_2 x^2 + a_4 x^4 +cdotsright]$$
The coefficients are then easily evaluated by inserting this in the differential equation and equating equal powers of $x$, the first few series coefficients are given by:
$$
begin{split}
a_2 &= -frac{1}{76}\
a_4 &= frac{547}{2945760}\
a_6 &=-frac{19079}{14104298880}
end{split}
$$
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
answered Jan 7 at 5:37
Count IblisCount Iblis
8,24121533
8,24121533
add a comment |
add a comment |
1
$begingroup$
Add a link for this video. I am not sure that this equation has an explicit solution.
$endgroup$
– Claude Leibovici
Jan 7 at 4:02
$begingroup$
@Claude Leibovici youtu.be/-_POEWfygmU He mentions it just at the end of the video. I’m guessing there isn’t an explicit solution because it’s non-linear. It’s deceptively simple looking so i though i’d give it a try lol.
$endgroup$
– Thatpotatoisaspy
Jan 7 at 4:16
2
$begingroup$
In general, nonlinear differential equations (like the one you are asking about) are hard to solve. The video seems to discuss only linear differential equations.
$endgroup$
– D.B.
Jan 7 at 4:22
1
$begingroup$
Even the time-homogeneous $ycdot y'' = -1$ looks nasty.
$endgroup$
– Henning Makholm
Jan 7 at 4:29
1
$begingroup$
@HenningMakholm I think with that one, you could do $$y'cdot y'' = -frac{y'}{y}$$and then integrate. That at least looks a bit solvable.
$endgroup$
– John Doe
Jan 7 at 5:05