Solve sine and exponential nonlinear differential equation?
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Is it possible to solve this kind of differential equation with forward Euler?
$$ddot y^2 + sin(ddot y ) + dot y + y = u$$
I haven't even write this ODE on the first order form. If I would do that, I would say $dot y = x_2$ and $y = x_1$. Then the ODE would be:
$$x_2 = dot x_1$$
$$dot x_2^2 + sin(dot x_2 ) + x_2 + x_1 = u$$
Then I move all the derivatives to the LHP and non-derivatives to RHP.
$$dot x_1 = x_2 $$
$$dot x_2^2 + sin(dot x_2 )= u - x_2 - x_1 $$
But how about these: $dot x_2^2 + sin(dot x_2 )$ ? My goal is to find $dot x_2$.
ordinary-differential-equations nonlinear-system non-linear-dynamics
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
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|
show 3 more comments
$begingroup$
Is it possible to solve this kind of differential equation with forward Euler?
$$ddot y^2 + sin(ddot y ) + dot y + y = u$$
I haven't even write this ODE on the first order form. If I would do that, I would say $dot y = x_2$ and $y = x_1$. Then the ODE would be:
$$x_2 = dot x_1$$
$$dot x_2^2 + sin(dot x_2 ) + x_2 + x_1 = u$$
Then I move all the derivatives to the LHP and non-derivatives to RHP.
$$dot x_1 = x_2 $$
$$dot x_2^2 + sin(dot x_2 )= u - x_2 - x_1 $$
But how about these: $dot x_2^2 + sin(dot x_2 )$ ? My goal is to find $dot x_2$.
ordinary-differential-equations nonlinear-system non-linear-dynamics
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
$endgroup$
$begingroup$
You can't find $dot{x}_2$ explicitly.
$endgroup$
– xpaul
Dec 19 '18 at 17:40
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And in many cases also not uniquely, especially if $2dot x_2+cos(dot x_2)=0$ or close to this point.
$endgroup$
– LutzL
Dec 19 '18 at 17:42
$begingroup$
@xpaul So the highest order need to be alone?
$endgroup$
– Daniel Mårtensson
Dec 19 '18 at 18:00
$begingroup$
$x^2 + sin(x)$ is not a one-to-one function of $x$.
$endgroup$
– Robert Israel
Dec 19 '18 at 18:25
$begingroup$
what is $u$ ? ...
$endgroup$
– JJacquelin
Dec 20 '18 at 7:19
|
show 3 more comments
$begingroup$
Is it possible to solve this kind of differential equation with forward Euler?
$$ddot y^2 + sin(ddot y ) + dot y + y = u$$
I haven't even write this ODE on the first order form. If I would do that, I would say $dot y = x_2$ and $y = x_1$. Then the ODE would be:
$$x_2 = dot x_1$$
$$dot x_2^2 + sin(dot x_2 ) + x_2 + x_1 = u$$
Then I move all the derivatives to the LHP and non-derivatives to RHP.
$$dot x_1 = x_2 $$
$$dot x_2^2 + sin(dot x_2 )= u - x_2 - x_1 $$
But how about these: $dot x_2^2 + sin(dot x_2 )$ ? My goal is to find $dot x_2$.
ordinary-differential-equations nonlinear-system non-linear-dynamics
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
$endgroup$
Is it possible to solve this kind of differential equation with forward Euler?
$$ddot y^2 + sin(ddot y ) + dot y + y = u$$
I haven't even write this ODE on the first order form. If I would do that, I would say $dot y = x_2$ and $y = x_1$. Then the ODE would be:
$$x_2 = dot x_1$$
$$dot x_2^2 + sin(dot x_2 ) + x_2 + x_1 = u$$
Then I move all the derivatives to the LHP and non-derivatives to RHP.
$$dot x_1 = x_2 $$
$$dot x_2^2 + sin(dot x_2 )= u - x_2 - x_1 $$
But how about these: $dot x_2^2 + sin(dot x_2 )$ ? My goal is to find $dot x_2$.
ordinary-differential-equations nonlinear-system non-linear-dynamics
ordinary-differential-equations nonlinear-system non-linear-dynamics
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
asked Dec 19 '18 at 17:33
Daniel MårtenssonDaniel Mårtensson
944416
944416
$begingroup$
You can't find $dot{x}_2$ explicitly.
$endgroup$
– xpaul
Dec 19 '18 at 17:40
$begingroup$
And in many cases also not uniquely, especially if $2dot x_2+cos(dot x_2)=0$ or close to this point.
$endgroup$
– LutzL
Dec 19 '18 at 17:42
$begingroup$
@xpaul So the highest order need to be alone?
$endgroup$
– Daniel Mårtensson
Dec 19 '18 at 18:00
$begingroup$
$x^2 + sin(x)$ is not a one-to-one function of $x$.
$endgroup$
– Robert Israel
Dec 19 '18 at 18:25
$begingroup$
what is $u$ ? ...
$endgroup$
– JJacquelin
Dec 20 '18 at 7:19
|
show 3 more comments
$begingroup$
You can't find $dot{x}_2$ explicitly.
$endgroup$
– xpaul
Dec 19 '18 at 17:40
$begingroup$
And in many cases also not uniquely, especially if $2dot x_2+cos(dot x_2)=0$ or close to this point.
$endgroup$
– LutzL
Dec 19 '18 at 17:42
$begingroup$
@xpaul So the highest order need to be alone?
$endgroup$
– Daniel Mårtensson
Dec 19 '18 at 18:00
$begingroup$
$x^2 + sin(x)$ is not a one-to-one function of $x$.
$endgroup$
– Robert Israel
Dec 19 '18 at 18:25
$begingroup$
what is $u$ ? ...
$endgroup$
– JJacquelin
Dec 20 '18 at 7:19
$begingroup$
You can't find $dot{x}_2$ explicitly.
$endgroup$
– xpaul
Dec 19 '18 at 17:40
$begingroup$
You can't find $dot{x}_2$ explicitly.
$endgroup$
– xpaul
Dec 19 '18 at 17:40
$begingroup$
And in many cases also not uniquely, especially if $2dot x_2+cos(dot x_2)=0$ or close to this point.
$endgroup$
– LutzL
Dec 19 '18 at 17:42
$begingroup$
And in many cases also not uniquely, especially if $2dot x_2+cos(dot x_2)=0$ or close to this point.
$endgroup$
– LutzL
Dec 19 '18 at 17:42
$begingroup$
@xpaul So the highest order need to be alone?
$endgroup$
– Daniel Mårtensson
Dec 19 '18 at 18:00
$begingroup$
@xpaul So the highest order need to be alone?
$endgroup$
– Daniel Mårtensson
Dec 19 '18 at 18:00
$begingroup$
$x^2 + sin(x)$ is not a one-to-one function of $x$.
$endgroup$
– Robert Israel
Dec 19 '18 at 18:25
$begingroup$
$x^2 + sin(x)$ is not a one-to-one function of $x$.
$endgroup$
– Robert Israel
Dec 19 '18 at 18:25
$begingroup$
what is $u$ ? ...
$endgroup$
– JJacquelin
Dec 20 '18 at 7:19
$begingroup$
what is $u$ ? ...
$endgroup$
– JJacquelin
Dec 20 '18 at 7:19
|
show 3 more comments
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$begingroup$
You can't find $dot{x}_2$ explicitly.
$endgroup$
– xpaul
Dec 19 '18 at 17:40
$begingroup$
And in many cases also not uniquely, especially if $2dot x_2+cos(dot x_2)=0$ or close to this point.
$endgroup$
– LutzL
Dec 19 '18 at 17:42
$begingroup$
@xpaul So the highest order need to be alone?
$endgroup$
– Daniel Mårtensson
Dec 19 '18 at 18:00
$begingroup$
$x^2 + sin(x)$ is not a one-to-one function of $x$.
$endgroup$
– Robert Israel
Dec 19 '18 at 18:25
$begingroup$
what is $u$ ? ...
$endgroup$
– JJacquelin
Dec 20 '18 at 7:19