- #1
Gyroscope
Homework Statement
[tex]-kx=m\frac{d^2x}{dt^2}[/tex]
I don't know how to solve differential equations, can someone show me how to do it, with this example.
Since both [itex]e^{i\omega t}[/itex] and [itex]e^{-i\omega t}[/itex] are solutions to the equation, then the general solution will be a linear combination of the two, with the constants determined by the boundary conditions.Why do you need both solutions?
Use the definition of complex exponential, namely [itex]e^{\pm i\theta}=\cos\theta \pm i\sin\theta[/itex]. With a bit of rearrangement, we find that [tex]\cos\theta =\frac{1}{2}(e^{i\theta}+e^{-i\theta})[/tex] and [tex]\sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})[/tex]Thanks cristo. How can you pass from e^(something) to cosine and sine functions?
While the method described above is very useful and practical, don't forget what the equation is asking:Homework Statement
[tex]-kx=m\frac{d^2x}{dt^2}[/tex]
I don't know how to solve differential equations, can someone show me how to do it, with this example.