# Understanding the equivalence of two general solutions of the harmonic oscillator differential equation

The solution of the following differential equation $$-kx(t) = m \frac{d^{2}x}{dt^{2}},$$ with $$\omega = \sqrt{k/m}$$, is $$x(t) = C_{1}e^{-i\omega t} + C_{2}e^{i\omega t}.$$ The real part of this, $$x(t) = x_0 e^{i(-\omega t + \phi)},$$ is the solution of my differential where $$x_0$$ is the amplitude of a harmonic oscillator and $$\phi$$ being a phase difference.

I have found another form of the general solution of this differential equation is $$x(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t).$$ With the following parameters $$x_0 = \sqrt{C_{1}^{2} + C_{2}^{2}}$$ , $$\cos\phi = C_1/x_0$$ and $$\sin\phi=-C_2/x_0$$ you can write $$x(t) = x_0\cos(\omega t + \phi),$$ which is equivalent of the $$\Re\{x_0e^{i(-\omega t + \phi)}\}$$. But I don't understand how the general solution of the differential equation can be written in these two ways.

• If your doubt concerns the equivalence between the two formulations, I would suggest to start from the latter $x_0 \cos{(\omega t + \phi)}$ and expand using the trigonometric identities (cosine of the sum). If your problem concerns the solution of the differential equation, I can write a more elaborate answer below. Let me know! Feb 9 at 0:55
• I am more curious about the different solutions of the differential equations but I guess I can see why they're equivalent but still unsure. Thank you an answer on this would be appreciated! Feb 9 at 1:15

You should not be using the same symbols $$C_1$$ and $$C_2$$ in the two solutions…
Write $$x(t)=D_1 e^{-i \omega t}+D_2 e^{i\omega t} \tag{1}$$ and expand using Euler’s formula: $$e^{i\omega t}=\cos(\omega t)+i\sin(\omega t)$$. Eq.(1) then takes the form $$x(t)=(D_1+D_2)\cos(\omega t) + i (D_1-D_2)\sin(\omega t) \tag{2}$$ so you can now declare $$C_1=D_1+D_2$$ and $$C_2=i (D_1-D_2)$$.
Note that here we must switch the physics on: obviously the position is real so $$D_1+D_2$$ must be real whereas $$D_1-D_2$$ must be pure imaginary.
This constrains the choices of $$D_1$$ to $$D_2$$. This is fine because the solution of Eq.(1) should depend on 2 real parameters, whereas it would depend on 4 real parameters if $$D_1$$ and $$D_2$$ were arbitrary complex numbers. In other words, $$D_1$$ and $$D_2$$ must be such hat $$D_1=D_2^*$$ so that then $$C_1$$ is related to the real part of $$D_1$$ and $$C_2$$ to the complex part of $$D_1$$.
You now start with Eq.(2) as $$x(t)=C_1\cos(\omega t)+C_2\sin(\omega t)$$ and then continue with $$C=\sqrt{C_1^2+C_2^2}$$, $$C_1=C\cos\varphi$$ and $$C_2=C\sin\varphi$$, which immediately gives $$x(t)=C \cos(\omega t+\varphi)$$