91Ó°ÊÓ

Given

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

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$a_n,a_n−1,…,a_1,a_0$as constants,

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

will have a characteristic equation of the form

$a_n{r}^n+a_{n−1}{r}^{n−1}+⋯+a_{1}r+a_0=0$

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

$y_{t+n}=b_1{y}_{t+n−1}+⋯+b_n{y}_t$

has characteristic equation

$r^n−b_1{r}^{n−1}−⋯−b_n=0$

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

$(D^2+{\mathrm{Ó\neg }}^2)I=0$

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

$\begin{array}{c}(D+i\mathrm{Ó\neg })(D−i\mathrm{Ó\neg })=0\\ I=0\\ D=Â\pm i\mathrm{Ó\neg }\end{array}$

Then solve the simpler equations $(D+i\mathrm{Ó\neg })I=0$and $(D−i\mathrm{Ó\neg })I=0$,

$\begin{array}{c}\frac{dI}{dt}=−i\mathrm{Ó\neg }\text{ }\\ I={\mathrm{Î\pm }}_1{e}^{−i\mathrm{Ó\neg }t}\\ \frac{dI}{dt}=i\mathrm{Ó\neg }\\ I={\mathrm{Î\pm }}_2{e}^{i\mathrm{Ó\neg }t}\end{array}$

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

$I={\mathrm{Î\pm }}_1{e}^{−i\mathrm{Ó\neg }t}+{\mathrm{Î\pm }}_2{e}^{i\mathrm{Ó\neg }t}$

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