Solving a second-order differential equation

[docnet]

Homework Statement

solve the equation with y = 1, y' = 0 at t = 0

Relevant Equations

y'' - ((y')^2)/y + (2(y')^2)/y^2 - ((y')^4)/y^4 = 0

Hi all, if anyone could help me solve this 2nd order differential equation, it would mean a lot.

Problem:

Solve the equation with y = 1, y' = 0 at t = 0

y'' - ((y')^2)/y + (2(y')^2)/y^2 - ((y')^4)/y^4 = 0

I have never solved an ODE of this kind before and I am not sure where to start. Wolfram and Symbolab calculators will not give an answer.

Thank you.

 

[Delta2]

Not sure if this will be helpful but check if the ansatz $$y=e^{kt}$$ is a solution for the proper values of $k$.
EDIT: NVM I was wrong, the ansatz doesn't satisfy y'(0)=0, unless we have $k=0$ but then the solution is trivial $y=1$. That is $y(t)=1$ is one solution for sure (you can easily verify it satisfies the differential equation given).
It would be interesting to see if for the moment we forget the initial condition y'(0)=0 and see what this ansatz gives if it is plugged to the ODE.
The idea behind this ansatz is that it satisfies $$(\frac{y'}{y})^n=k^n$$ and also that $$y''=k^2y$$ so the ODE might be turned into an algebraic equation for k.

 

[epenguin]

I'd say that the only way to solve a differential equation, at least the only way that is commonly taught, is to know the answer! Or at least to know the sort of answer for things like it. And then with the particular case you hammer it into the shape that fits (and don't forget to check the answer which is very simple to do but often forgotten).

And then there is the Polya principle "have you ever seen anything like it?". For instance there is a trick that comes up often in dynamics - what is the derivative of (yy')? I haven't got the answer but I would play around with that, you do have to play around a bit.

 

[haruspex]

then the solution is trivial y=1.

Quite so. What makes you think there are others?

 

[Delta2]

Quite so. What makes you think there are others?

I don't know to be honest at start I thought for some reason it was linear (which isn't of course) and we know linears of order n have at least n linearly independent solutions but it isn't linear so i don't know

If we relax the initial conditions then I think one solution is $$y=ce^{\pm\sqrt{2}t}$$

 

[aheight]

and I am not sure where to start. Wolfram and Symbolab calculators will not give an answer.

Thank you.

Equations of the form $f(y,y',y'')=0$ can be reduced to a first-order in y and y' by letting $y'=p$ so that $y''=p\frac{dp}{dy}$. Maybe see if this helps. Also look up "independent variable missing" in DE text for some additional info. Wouldn't hurt too to just solve it numerically via NDSolve in Mathematica to at least see what it looks like.

 

[phyzguy]

At t=0, y=1 and y'=0. Plugging into the equation also gives that y''=0 at t=0. If y'(0)=0 and y''(0)=0, then what does that tell you?

 

[haruspex]

At t=0, y=1 and y'=0. Plugging into the equation also gives that y''=0 at t=0. If y'(0)=0 and y''(0)=0, then what does that tell you?

Yes, my hint in post #4 was probably too subtle.

 

[docnet]

I am sorry all, my original post included a bad typo in the equation. I am not experienced at typing equations into the computer.

Here is the entire question and my work leading up to the differential equation in part (f.) The professor warned us ahead of time the ODE for the geodesic may be difficult.

I believe the last line is the correct ODE to solve for, which I was unable to solve using any methods I was familiar with. Looking at the plot of the geodesic, it looks like the V component asymptotically approaches 1 and -1 at infinite and -infinite t. the U component only exists for > 0 and is also symmetric around t = 0. Does anyone know of any similar ODE examples to learn from?

 

[phyzguy]

I think the best way to proceed is to use the hint from @aheight in post #6, and replace u'' with u' du'/du. This at least converts the equation to a first order equation.

 

[haruspex]

I think the best way to proceed is to use the hint from @aheight in post #6, and replace u'' with u' du'/du. This at least converts the equation to a first order equation.

... and beyond that, u'=u cos(θ) looks useful.

 

[docnet]

... and beyond that, u'=u cos(θ) looks useful.

Hm... I tried solving the expression u' = u cos(Θ) and obtained u = C exp [sin(Θ) ] which doesn't seem to correspond to a solution for our problem.

 

[haruspex]

Hm... I tried solving the expression u' = u cos(Θ) and obtained u = C exp [sin(Θ) ] which doesn't seem to correspond to a solution for our problem.

No, I mean after replacing u'' with u' du'/du, substitute for u' using that relationship to get an ODE in x and θ.

 

[docnet]

After doing the substitutions we ended up with the following equation

where t = θ and u = y.

The 1st order ODE could not be solved by using Symbolab. Is there a mistake??

 

[haruspex]

So we substitute u' = ucosΘ

and substitute u = (1/2)u²cosΘ obtained by integrating u'

and substitute du'/du = cosΘ obtained by differentiating u'

and ended up with the following o^th order equation that cannot be solved for u.

https://www.physicsforums.com/attachments/274852

Should we substitute u' = ucosΘ and leave u and du'/du alone?

edit: the expression u = (1/2)u²cosΘ does not make any sense. I'll spend some time before posting a followup.

You have $\ddot u=\frac{\dot u^2}u+\frac 1{u^2}(1-\frac{\dot u}u)^2$.
Writing $x=u, y=\dot u, y'=\frac{dy}{dx}, \ddot u= \frac d{dt}y=\frac d{du}y\frac{du}{dt}=yy'$:
$x^2yy'=xy^2+(1-\frac{y^2}{x^2})^2$
If all that looks right to you, substitute $y=x\cos(\theta)$ and see what simplifies.

 

[docnet]

You have $\ddot u=\frac{\dot u^2}u+\frac 1{u^2}(1-\frac{\dot u}u)^2$.
Writing $x=u, y=\dot u, y'=\frac{dy}{dx}, \ddot u= \frac d{dt}y=\frac d{du}y\frac{du}{dt}=yy'$:
$x^2yy'=xy^2+(1-\frac{y^2}{x^2})^2$
If all that looks right to you, substitute $y=x\cos(\theta)$ and see what simplifies.

So we have

which simplifies to

then

which gives y = x in our substitution

and

with the solution

and

with the solution

with c1= 1 and c = 0.

May I continue solving the next part of the problem?

 

[haruspex]

So we have
View attachment 274854

No, θ is a variable, so what does y' become?

 

[docnet]

No, θ is a variable, so what does y' become?

Differentiating y with respect to x gives us y'=cosθ which gives

View attachment 274854

The first two terms cancel and we are left with the equation in θ as before.

when we differentiate y with respect to theta we have y' = - xsinθ which gives us an equation which is difficult to solve for x or θ.

We could differentiate with respect to x and theta using the chain rule, and it results in another equation which I was unable to solve for x or θ.

 

[haruspex]

Differentiating y with respect to x gives us y'=cosθ which gives

No it doesn’t. θ is a variable, as in θ=θ(x).

 

[docnet]

I'm getting

which simplies to

whose solution is Θ(x) = 0

May I move onto the next part?

 

[haruspex]

I'm getting

View attachment 274915

You have made a mistake in expanding $(1-\frac{y^2}{x^2})^2$.

 

[docnet]

You have made a mistake in expanding (1−y2x2)2.

Doh! yes I did, and I just fixed my mistake.

It finally comes out to

whose solutions are

where y(t) has been replaced with Θ(x) and t with x.

 

[haruspex]

Doh! yes I did, and I just fixed my mistake.

It finally comes out to

View attachment 274923

Which can be simplified to $x^4\sin(\theta)\cos(\theta)+\sin^4(\theta)=0$, hence $x^4\cos(\theta)+\sin^3(\theta)=0$, unless sin() is 0.

 

[haruspex]

whose solutions are

View attachment 274922

where y(t) has been replaced with Θ(x) and t with x.

Defining a different y from what we had earlier is confusing.
At t=0, u=x=1, $y=\dot u=0$, $\cos(\theta)=0, |\csc(\theta)|=1$.

I get $(\dot u)^2=\frac{2u^5-2u^2}{5u^3-2}$, leading to a nasty-looking integral.

 

[docnet]

Is anyone familiar with the surface corresponding to our metric

?



It could give us an idea of what the geodesic is.

It looks like the surface stretches the du vector to infinity at u = 0 and shortens the dv vector to 0 at u = 0.

 

[docnet]

Dear all,

There is another error in our differential equation.

Starting over from part e) we solve the following equation for (dv/dt)².

giving

and substitute it in the following equation

giving

Which offers no clear way to solve. We examine the following expression from part d).

which makes

which implies

for some constant c. solving,

substitute into the first equation to get

which makes

I think this way is not easy.

 

[docnet]

The expression

has the solution

and

 

[docnet]

For the v component of the geodesic, we plug in

into the unit speed equation

giving

and

We then integrate to give

and

which is incorrect judging from the following plot in the uv plane

but I think we are getting closer to the correct solution.

 

[haruspex]

For the v component of the geodesic, we plug in
View attachment 275735

into the unit speed equation
View attachment 275736

giving
View attachment 275731

You seem to have squared part of $\frac{du}{dt}$ twice. The left hand term above should reduce to 1/2.

 

[dsaun777]

What course is this? Differential geometry?

 

[docnet]

You seem to have squared part of $\frac{du}{dt}$ twice. The left hand term above should reduce to 1/2.

I am sorry I should apologize because the solution to our ODE is not a geodesic, because we are solving the wrong ODE.

Not solving this problem has been eating me up. So, I revisited our previous calculations tonight, made sure to be extra careful not to make errors and ended up with the following. I still have not learned how to write LaTex equations, so these were typed in Microsoft Word, and these will not show up in quoted messages.

I am unable to solve this nonlinear second order ODE by any method I am familiar with.

Is anyone able to compute it using Mathematica or Wolfram Αlpha?

Thank you.

 

[docnet]

Acccording to DSolve in Mathematica, the fearful ODE $$\frac{d^2u}{dt^2}=\frac{1}{u^3}-\Big(\frac{u^4+1}{u^5}\Big)\Big(\frac{du}{dt}\Big)^2$$
has "closed form" solutions given by...

 

[docnet]

By using DSolve with initial value conditions,

$$\frac{d^2u}{dt^2}=\frac{1}{u^3}-\Big(\frac{u^4+1}{u^5}\Big)\Big(\frac{du}{dt}\Big)^2\Rightarrow u(t)=\sqrt{1+t^2}$$

 

[docnet]

(g) we plug $u(t)$ into the unit speed equation to find
$$\frac{dv}{dt}=\Big(\frac{t^4+t^2+1}{(1+t^2)^3}\Big)^{\frac{1}{2}}$$

which is a separable equation, when trying to compute
$$v(t)=\int \Big(\frac{t^4+t^2+1}{(1+t^2)^3}\Big)^{\frac{1}{2}}dt$$

we cannot find an explicit closed form solution