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Second order

I was trying to solve a physics problem which led me to an equation of the form:
d^2x/dt^2 + a/ x = b; Can this be solved without any aproximations beeing made?
 

Answers and Replies

Depends.

Assuming a,b are constants, multiply by [tex]x^{\prime}[/tex], to get

[tex]x^{\prime} x^{\prime \prime} + a \frac{x^{\prime}}{x} = b x^{\prime}[/tex]

now integrate to get

[tex]\frac{x^{\prime}^2}{2} + a \ln{x} = bx + c [/tex]

(c is a constant). Rearrange to get t as a function of x, i.e.,

[tex]\frac{1}{\sqrt{2}} \int{\frac{dx}{\sqrt{bx + c - a \ln{x}}}} = t + const.[/tex]

Other than that, I'm not sure.
 

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