# Derivation of Solution to nonlinear 2nd Order ODE

## Main Question or Discussion Point

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.

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HallsofIvy
Homework Helper
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
Homework Helper
Gold Member
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.

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?

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.

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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.