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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


             ole.gif


Solution. The general solution is given by


             ole1.gif


Proof



II Equations of type


             ole2.gif


Solution. This equation can be written as


1)        dp/dx = f(p)


where p = dy/dx. Rewriting 1) we get


ole3.gif


Integrating, we have


ole4.gif


Now if 3) can be solved for p to give


            p = g(x)


the solution will be given by


             ole5.gif


 


III Equations of type


             ole6.gif


(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


             ole7.gif




IV Equations of type


             ole8.gif


(independent variable x missing).



Solution. This equation can be written as


4)        f(y, p, dp/dx) = 0


where p = dy/dx. Noting that


             ole9.gif


let us substitute p(dp/dy) for dp/dx in 4) to give


             ole10.gif

 

If this equation can be solved for p to give


            p = g(y)


the solution will be given by


             ole11.gif




References

1. Eshbach. Handbook of Engineering Fundamentals.



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