Please help Need solution to Second Order nonlinear DE

[ELEN_guy]

Does anyone know how to solve the following Non-linear, second order, differential equation?

A*y" + B*(y')^2 = F(t) + C

where A, B, & C are constants

**please note, in case the above notation isn't clear, the y' term is squared which is what makes it non-linear. Also, F(t) is time dependent.

I tried using the following substitution:

y' = u ..giving rise to.. y" = u'

this yields the following DE:

A*u' + B*u^2 = F(t) + C

which is now at least a first-order DE but I still can't solve it.

does anyone knows how to solve either one of these DE's please let me know.

Thanks^googol

 

[flatmaster]

Is your differeation variable of your unknown function also t?

y = y(t) and y' = dy/dt


If not than F(t) + C can be called some new constant.

 

[flatmaster]

Are A, B, and C necessarily independent?

 

[ELEN_guy]

The differentiation variable is NOT t, its just some other variable..lets say x:

y = y(x), y' = dy/dx..and so on

yeah that makes sense.. F(t) + C = K

A, B, and C are constants which are not zero, one, or equal to each other.

 

[flatmaster]

A*y" + B*(y')^2 = K

I think it might be a sum of two different powers of x. Try a power series.


inf
y = Sum {Dn x^n }
n=0

I'll try on paper.

You know how people get thoes nice emeded forumlas that look like mathmatica? Is that imbeded in this forum somewhere?

 

[ELEN_guy]

Thanks! not sure how to use power series to solve DE's...never covered that topic when I took my diff eq course. don't know how to imbed those formulas either..

 

[flatmaster]

What does this DE physically represent or is this just a math exercise?

 

[ELEN_guy]

yes, it represents the equation of motion of an inertial system with a drag force and an external force which is NOT time-related. Namely, its an idealized model of the Apollo reentry.

The differentiation variable is, in fact, time dependent while the F(x) + C term is not...I just wrote the reverse since it was easier to type in using "primes" instead of "dots" without the imbedded notation.

I appreciate your help!

 

[flatmaster]

Well, I'm on a library computer and I just discovered they have Maple (which I don't know well). Without knowing the simplify function, this is what maple spit out.

y(x) = -(1/2)*(2*sqrt(k)*sqrt(b)*x+ln(4*k/(b*(_C1*exp(2*sqrt(k)*sqrt(b)*x/a)-_C2)^2))*a)/b

I'm trying to find the FullSImplify[] function. C1 and C2 are the two constants required for a 2nd degree DE. That expression's not much use unless you have something to clean it up on your side.

 

[ELEN_guy]

Once again, thanks a lot for taking the time to help! I totally forgot about Maple..I'm going to see if anyone on my team knows it well. At this point I'm about ready to take that expression and make up some constants to fit the data.

 

[flatmaster]

Apollo re-entry. Cool problem.

A*y" + B*(y')^2 = F(x) + C

You say the primes are actually time derivitives. I assume the y'' was the acceleration term and the (y')^2 term is the drag force, making the actual function the height of the craft as a function of time?

Wouldn't drag force also be a function of the density of the atmosphere, thus making the drag term some function of your height y?

What are the other terms?

 

[HallsofIvy]

Since only the derivatives of y appear in that equation, the obvious thing to do is to let v= y' and have Av'+ Bv²= F(t)+ C a first order equation. v'= (F(t)+ C Bv²)/A. How you would solve that would depend strongly on the form of F(t).

That's assuming that the differentiation is with respect to t. If it is with respect to x:

The differentiation variable is NOT t, its just some other variable..lets say x

it's just dv/(F(t)+ C- Bv²)= Adx which is easy to integrate for each t.