When applying Kirchoff's voltage law to a $\text{LCR}$ series circuit. The following differential equation pops up..
$$E\sin(Kt)-\dfrac{q}{C}-R\dfrac{dq}{dt}-L\dfrac{d^2q}{dt^2}=0$$
Where
$E$, $K$, $R$, $C$ and $L$ are positive constants.
I tried solving the equation by assuming the solution to be of the form $A\sin(Bx)+C\cos(Dx)$ and then solving for the constants but it didn't work.
I'm in 12th grade and only know how to solve differential equations like linear first degree.
Best Answer
Don't know if you are acquainted with complex numbers.
These type of equations are easily solved (electrical engineers know very well) in the complex field.
Pass from $E\sin(Kt)$ to $V=E e^{iKt}$, and also express $q=Q e^{iK(t+\tau)}$, or better just $q=Q e^{iKt}$, allowing $Q$ to be complex .
To take the derivatives is easy, and you arrive to a complex equation to be solved for $Q$ and which is linear in $V$ ($V$ divided by a complex expression in $R,L,C$).
In that, take the immaginary component of both sides, and that's all.