Find (in terms of$L$and$C$) the frequency of electrical oscillations in a series circuit (Figureif$\mathbf{R}=\mathbf{0}$or$V=0$, but$I\ne 0$. (When you tune a radio, you are adjusting$C$and/or$L$to make this frequency equal to that of the radio station.)

Given

$\begin{array}{l}R=0\\ V=0\\ I\ne 0\end{array}$

Auxiliary equation:

Auxiliary equation is an algebraic equation of degree$n$upon which depends the solution of a given$n$th-order differential equation or difference equation. The auxiliary equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with$y$as the dependent variable, superscript () denotingth-derivative, and$a_n,a_n-1,\dots ,a_1,a_0$as constants,

$a_n{y}^{\left(n\right)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\text{'}}+a_{0}y=0$

will have a characteristic equation of the form

$a_n{r}^n+a_{n-1}{r}^{n-1}+\cdots +a_{1}r+a_0=0$

whose solutions$r_1,r_2,\dots ,r_n$are the roots from which the general solution can be formed. A linear difference equation of the form

$y_{t+n}=b_1{y}_{t+n-1}+\cdots +b_n{y}_t$

has characteristic equation

$r^n-b_1{r}^{n-1}-\cdots -b_n=0$

When R=V=0

$\begin{array}{c}L\frac{d^{2}I}{d{t}^2}+\frac{I}{C}=0\\ \frac{d^{2}I}{d{t}^2}+\frac{L}{C}I=0\end{array}$

which is a second order differential equation.

solve this differential equation starting by write the auxiliary equation

$(D^2+{\omega }^2)I=0$

Here $D$is the differential operator, and. ${\omega }^2=1/LC$Differential equation is, start by find the roots of the auxiliary equation

$\begin{array}{c}(D+i\omega )(D-i\omega )=0\\ I=0\\ D=\pm i\omega \end{array}$

Then solve the simpler equations $(D+i\omega )I=0$and $(D-i\omega )I=0$,

$\begin{array}{c}\frac{dI}{dt}=-i\omega \\ I={\alpha }_1{e}^{-i\omega t}\\ \frac{dI}{dt}=i\omega \\ I={\alpha }_2{e}^{i\omega t}\end{array}$

and the linear combination of these two solutions is the general solution for our differential equation

$I={\alpha }_1{e}^{-i\omega t}+{\alpha }_2{e}^{i\omega t}$

Notice that,write the general solution in other forms. Conclude that the frequency of the electrical oscillations in such circuit is$\omega =\frac{1}{\sqrt{LC}}$