- #1
sirpsycho85
- 7
- 0
This isn't homework, but it feels like it...it's from the Dorf & Svoboda Introduction to Electric Circuits, 6th ed. on p 298, if you happen to have that.
I'm following the derivation of the formula for capacitor voltage for an RC circuit with a single resistor and single cap, or the current in the simple inductor circuit. Arriving at the general form of the differential equation, with time constant T:
dx(t)/dt + x(t)/T = K
rewritten:
dx/dt = (KT - x)/T
now, in the next step, the authors separate the variables and multiply each side by -1, yielding:
dx/(x - KT) = -dt/T
and go on to integrate both sides and arrive at:
ln(x - KT) = -t/T + D
D being the constant of integration. Raising e to both sides you get:
x(t) = KT + Ae-t/T
where A is eD, and you can go on using initial conditions to solve for the constant.
I can't get the same result when I don't multiply both sides by -1 when separating variables before solving the equation. My steps are as follows:
dx/dt = (KT - x)/T
dx/(KT - x) = dt/T having not multiplied by -1
ln(KT - x) = t/T + D
KT - x = Aet/T
x(t) = KT - Aet/T
This doesn't appear to be the same solution, as en is not equal to -e-n.
Can somebody please help me figure out what I've missed?
Oh, and hello everybody. I'm a junior engineer but work in a field that barely ever touches on a lot of what I learned in school...I've decided to start studying again to make sure I retain the fundamentals, especially if I do switch into a hardware design position.
Thank you.
I'm following the derivation of the formula for capacitor voltage for an RC circuit with a single resistor and single cap, or the current in the simple inductor circuit. Arriving at the general form of the differential equation, with time constant T:
dx(t)/dt + x(t)/T = K
rewritten:
dx/dt = (KT - x)/T
now, in the next step, the authors separate the variables and multiply each side by -1, yielding:
dx/(x - KT) = -dt/T
and go on to integrate both sides and arrive at:
ln(x - KT) = -t/T + D
D being the constant of integration. Raising e to both sides you get:
x(t) = KT + Ae-t/T
where A is eD, and you can go on using initial conditions to solve for the constant.
I can't get the same result when I don't multiply both sides by -1 when separating variables before solving the equation. My steps are as follows:
dx/dt = (KT - x)/T
dx/(KT - x) = dt/T having not multiplied by -1
ln(KT - x) = t/T + D
KT - x = Aet/T
x(t) = KT - Aet/T
This doesn't appear to be the same solution, as en is not equal to -e-n.
Can somebody please help me figure out what I've missed?
Oh, and hello everybody. I'm a junior engineer but work in a field that barely ever touches on a lot of what I learned in school...I've decided to start studying again to make sure I retain the fundamentals, especially if I do switch into a hardware design position.
Thank you.