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Second order differential equations
There are general procedures for solving some special second order equations. We list some:
I Equations of type
Solution. The general solution is given by
II Equations of type
Solution. This equation can be written as
1) dp/dx = f(p)
where p = dy/dx. Rewriting 1) we get
Integrating, we have
Now if 3) can be solved for p to give
p = g(x)
the solution will be given by
III Equations of type
(dependent variable y missing).
Solution. This equation can be written as
f(x, p, dp/dx) = 0
where p = dy/dx. If this equation can be solved for p to give
p = g(x)
the solution will be given by
IV Equations of type
(independent variable x missing).
Solution. This equation can be written as
4) f(y, p, dp/dx) = 0
where p = dy/dx. Noting that
let us substitute p(dp/dy) for dp/dx in 4) to give
If this equation can be solved for p to give
p = g(y)
the solution will be given by
References
1. Eshbach. Handbook of Engineering Fundamentals.