Integrating a second derivative-involving solution for Simple Harmonic Motion

[insertnamehere]

Hello, i am now in the process of integrating m(d^2x/dt^2)=-kx which i know i will have to do twice in order to obtain the general solution to simple harmonic motion, x= Acos(wt+c) c=phi

but I'm just having problems with the second derivative of acceleration (d^2*x/dt^2) when it comes to integrating, I tried separating them into dv/dt and then dv/dx and dx/dt, so i obtain v, etc... but this complicates things even more for me! It would be so great if someone could give me a hint as to where i can start in integrating with this second derivative...
Thank you very much for your time.

 

[Fermat]

Hello, i am now in the process of integrating m(d^2x/dt^2)=-kx which i know i will have to do twice in order to obtain the general solution to simple harmonic motion, x= Acos(wt+c) c=phi

but I'm just having problems with the second derivative of acceleration (d^2*x/dt^2) when it comes to integrating, I tried separating them into dv/dt and then dv/dx and dx/dt, so i obtain v, etc... but this complicates things even more for me! It would be so great if someone could give me a hint as to where i can start in integrating with this second derivative...
Thank you very much for your time.

The 2nd differential is often rearranged like this,

$$\frac{d^2x}{dt^2}= v\frac{dv}{dx}$$

and you get that this way,

$$\frac{d^2x}{dt^2}=\frac{d}{dt}(dx/dt)=\frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt}= \frac{dv}{dx}v = v\frac{dv}{dx}$$

substitute for $$\frac{d^2x}{dt^2}$$ to $$v\frac{dv}{dx}$$ then do the integration.

 

[insertnamehere]

Yes, i tried doing that, but then eventually i get stuck with
(v^2)/2= (-k/m)(x^2)/2 + C
Would i have to integrate again to obtain the general solution of
x=Acos(wt+c)? What can i do with the v^2 and the x^2, and the negative k/m?

 

[Fermat]

Your halfway there.
From the initial conditions of the problem, work out a value for C.
rewrite the eqn as v = whatever ...
convert v to dx/dt and integrate agan.