Solution strange, differential equation

[Phizyk]

Hi!
I have big problem with solve this equation:
$$m\frac{d^{2}x}{dt^{2}}+ksinx=0$$
I can't go ahead, because I don't know how solve this
$$\frac{dx}{\sqrt{cosx}}=\sqrt{\frac{2k}{m}}dt$$
Phizyk

 

[HallsofIvy]

Since the independent variable, t, does not appear explicitely in the equation, that is a candidate for "quadrature".

Let v= dx/dt. Then d²x/dt²= dv/dt. But by the chain rule, dv/dt= (dv/dx)(dx/dt). And dx/dt= dv/dt, of course. That is d²x/dt²= (dx/dt)(dv/dt)= vdv/dx.

Your differential equation can be reduced to vdv/dx= -ksin(x) which is "separable":
mvdv= - k sin(x)dx. Integrating both sides, (m/2)v²= k cos(x)+ C. (That square is the reason for the name "quadrature".) Then v²= (2k/m) cos(x)+ C' or
v= dx/dt= sqrt((2k/m) cos(x)+ C').

I assume that, in order to get rid of that constant of integration, C', you must have some initial condition on dx/dt.

 

[Phizyk]

Great. Thanks Hallsoflvy.

 

[Phizyk]

But this equation $$\frac{dx}{dt}=\sqrt{\frac{2k}{m}cosx+C^{'}}$$ can I solve? Can I obtain x(t)? For t=0 x=0. It's a equation of motion.