Can anyone solve this integral for spring mass oscillation?

[jin94]

Homework Statement

So this is the question I need to solve. I was told to directly integrate the bottom equation to solve for x(t). I can solve (1) by differentiating (1) to turn it into 2nd order difffeq (mx''=kx), but I think I'm not allowed to do that ... I have no idea how to solve (2) using direct integration. It looks like a difficult non-linear equation to solve to me. please help me T T

Homework Equations

The Attempt at a Solution

I solved (1) by turning into mx''=kx, but I'm not allowed to do that.

 

[TSny]

Welcome to PF!

Equation (2) is in terms of the two variables x and t. The ideas is to "separate" the two variables so that just the x variable occurs on one side and the t variable on the other, in such a way that the equation "begs" you to integrate both sides.

 

[jin94]

Welcome to PF!

Equation (2) is in terms of the two variables x and t. The ideas is to "separate" the two variables so that just the x variable occurs on one side and the t variable on the other, in such a way that the equation "begs" you to integrate both sides.

Thank you! but then how do I integrate 1/(a-b*x^2)^(1/2) dx?

 

[TSny]

This is a standard integral that can be done with a trig substitution. Or an integral table
http://integral-table.com/downloads/single-page-integral-table.pdf

 

[jin94]

This is a standard integral that can be done with a trig substitution. Or an integral table
http://integral-table.com/downloads/single-page-integral-table.pdf

Thank you so much!