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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.

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