Derivation of Solution to nonlinear 2nd Order ODE

[X89codered89X]

I need to derive the solution for the differential equation analytically:

y'' + g(t,y(t)) = 0
y'(0) = z_o
y(0) = y_o

I know the solution is:

y(t) = y_o + z_ot - single integral from 0 to t of (t-s)g(s,y(s))ds

I believe I need to assume something about the solution being a function of e^at somehow due to no damping, but I'm not sure.

 

[HallsofIvy]

I need to derive the solution for the differential equation analytically:

y'' + g(t,y(t)) = 0
y'(0) = z_o
y(0) = y_o

I know the solution is:

y(t) = y_o + z_ot - single integral from 0 to t of (t-s)g(s,y(s))ds

I believe I need to assume something about the solution being a function of e^at somehow due to no damping, but I'm not sure.

What you give is NOT a solution because it involves the unknown function, y, in the integral. It is, rather, an integral equation equivalent to the given differential equation.

 

[LCKurtz]

And to add to what Halls said, there is no hope of solving your equation analytically in the general case. With some specific functions g it is possible.

 

[X89codered89X]

Well, Halls yes. I suppose I'm not supposed to find a specific solution to the differential equation. But i do need to derive that equivalent integral equation. Can you help?

 

[jackmell]

I need to derive the solution for the differential equation analytically:

y'' + g(t,y(t)) = 0
y'(0) = z_o
y(0) = y_o

I know the solution is:

y(t) = y_o + z_ot - single integral from 0 to t of (t-s)g(s,y(s))ds

I believe I need to assume something about the solution being a function of e^at somehow due to no damping, but I'm not sure.

You don't need to assume that. But you do need to know how to convert a differential equation to an equivalent Volterra integral equation. Find a book on intro to integral equations. First note the transformational formula (for derivation and n'th case, see integral equation text):

$$\int_0^t \int_0^t f(t)dtdt=\int_0^t(t-s)f(s)ds$$

Now let:

$$y''=-f(t,y)$$

and integrate both sides from 0 to t:

$$\int_0^t y'' dt=-\int_0^t f(t,y)dt$$

$$y'(t)-z0=-\int_0^t f(t,y)dt$$

Now integrate again and use the transformational expression to arive at the Volterra integral equation.

 

[X89codered89X]

I just got the homework back. I used to volterra transformation but I was basically supposed to derive the transformation myself without just using it, which is pretty much what I expected. I had the prof explain to me how fundamentally it was possible to change a double integration into a single integration. He drew on the board on how to do it, change of variables using basically the technique I also found outlined here on wikipedia:

http://en.wikipedia.org/wiki/Order_of_integration_(calculus)#Relation_to_integration_by_parts

Even from reading this section, for some reason I'm still struggling a ton to get it.