Editing LC circuit (section)

=== Solution ===
Thus, the complete solution to the differential equation is

: <math>I(t) = Ae^{+j \omega_0 t} + Be^{-j \omega_0 t}</math>

and can be solved for {{mvar|A}} and {{mvar|B}} by considering the initial conditions. Since the exponential is [[complex numbers|complex]], the solution represents a sinusoidal [[alternating current]]. Since the electric current {{mvar|I}} is a physical quantity, it must be real-valued. As a result, it can be shown that the constants {{mvar|A}} and {{mvar|B}} must be [[complex conjugate]]s:

: <math>A = B^*.</math>

Now let

: <math>A = \frac{I_0}{2} e^{+j \phi}.</math>

Therefore,

: <math>B = \frac{I_0}{2} e^{-j \phi}.</math>

Next, we can use [[Euler's formula]] to obtain a real [[Sine wave|sinusoid]] with [[amplitude]] {{math|''I''<sub>0</sub>}}, [[angular frequency]] {{math|''ω''<sub>0</sub> {{=}} {{sfrac|1|{{sqrt|''LC''}}}}}}, and [[Phase (waves)|phase angle]] <math>\phi</math>.

Thus, the resulting solution becomes

: <math>I(t) = I_0 \cos\left(\omega_0 t + \phi \right),</math>

: <math>V_L(t) = L \frac{\mathrm{d}I}{\mathrm{d}t} = -\omega_0 L I_0 \sin\left(\omega_0 t + \phi \right).</math>