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In this video we're going to examine the solution to a second-order system I find that when you're first starting out solving these it's very useful to create a little cheat sheet to kind of help you keep all of the information that you need about second-order differential equations in one convenient spot this is the cheat sheet that I've created and you can pause the video and take a look at it if you use it to solve a dozen or more problems you'll find that a lot of this stuff you kind of end up memorizing just from using it so frequently we'll come back to it throughout the video okay so here we have a circuit and we know that it's a second-order system we know that it's a second-order circuit because it has two energy storage devices okay and we're further tipped off to this being a circuit that's going to require some sort of differential equation because of the quantum event when the switch is flipped open here so our ultimate solution is going to be to find an equation that describes the current I of T at all points greater than T equals zero when you're solving these circuits this is the algorithm that I use first I'll determine the initial conditions the initial conditions are the current through the inductor at T equals zero the voltage across the capacitor at T equals zero the variable that you're solving for in this case I of T at T equals zero plus and the the terminal value of that parameter that you're looking for so I at infinity notice that I don't care about whether I'm saying I the current through the inductor at zero minus or zero plus or the voltage across the capacitor at zero minus or zero plus because the energy in a capacitor and inductor cannot change instantaneously so those values will be the same before the quantum event and just after the quantum event ii will write an equation to describe the circuit if we have a second-order circuit like this one we are guaranteed to have a second-order differential equation we're going to write that differential equation in standard form and use that to extract the characteristic equation the characteristic equation will give us all of these variables we'll use these variables to solve the rest of the circuit and the case is going to give us the general equation in my cheat sheet I have a general equation written for each of the cases here it is here here it is here so on and so forth we're going to use that general equation and the derivative of that general equation to solve the circuit and it works like this the general we'll be able to solve the general equation with one of those initial conditions that's on the energy one of the energy storage device then we'll be able to use the derivative of the general.