Solve DE using integ factor, please check work thanks

[darryw]

Homework Statement

R dQ/dt + Q/C = 0 ...Q(0) = Q_0

dQ/dt = - Q/RC

integrating factor: mu(x) = e^integ(1/RC)
mu(x) = e^(t/RC)

(e^(t/RC) Q)' = integ 0

e^(t/RC) Q = t + c

Q = te^-(t/RC) + ce^-(t/RC)

then apply initial conditions of Q(0) = Q_0

Q_0 = c

the last part seems sort of weird to me?? thanks

Homework Equations

The Attempt at a Solution

 

[Cyosis]

By simply plugging your solution into the differential equation you would've noticed that your solution is wrong. Why are you using an integrating factor on a separable differential equation?

(e^(t/RC) Q)' = integ 0

e^(t/RC) Q = t + c

integ 0 != t+c

 

[darryw]

yes i noticed it can be solved as separable, but shouldn't it give same answer using integrating factor?

my mistake was integral of zero is not t, it is just the arb constant, c
so...

(e^(t/RC) Q)' = integ 0

e^(t/RC) Q = c

Q = c^e^-(t/RC)

is this correct (so far) ? thanks

 

[Cyosis]

You have raised the constant to the power of the exponent. I reckon that is a typo? If it is a typo then yes your answer is correct.

 

[darryw]

thanks.. yup, typo..should be..
Q = ce^-(t/RC)

how exactly do i apply the init cond though?
it says Q(0) = Q_0, so doesn't that mean:

Q_0 = ce^-(0/RC)

Q_0 = c ?

this is what confuses me, because i am used to initial conditions being something like y(0) = 1, for example, then i just plug in the y value and x value to solve for the constant.. but when it says Q(0) = Q_0, i don't know what the problem is asking??
thanks.

 

[Cyosis]

You should think of it like this. You have a solution Q(t) and an initial condition Q(0)=Q_0. This tells you that your general solution Q(t) must be equal to Q_0 when t=0. The only way for your general solution to meet that requirement is to set c=Q_0 as you already did.

 

[darryw]

ok i think i got it.
so when init conditions, answer is :
Q = Q_0e^-(t/RC)

So if i solve as separable, i get this...R dQ/dt + Q/C = 0 ...Q(0) = Q_0

dQ/dt = -Q/C

(1/Q)dQ = -(1/C)dt

then integrate both sides...

ln|Q| = -ln|C|+c

exponentiate both sides...

Q = -C + e^c

this last part looks wrong though.. ??
thanks

 

[Cyosis]

ok i think i got it.
so when init conditions, answer is :
Q = Q_0e^-(t/RC)

Correct.

R dQ/dt + Q/C = 0 ...Q(0) = Q_0

dQ/dt = -Q/C

(1/Q)dQ = -(1/C)dt

then integrate both sides...

ln|Q| = -ln|C|+c

You forgot the R in your first step.

Your second step is correct if you add the R.

In the 4th step you have integrated over C, but C is a constant. You need to integrate with respect to t.

 

[darryw]

R dQ/dt + Q/C = 0 ...Q(0) = Q
dQ/dt = -Q/RC

(1/Q)dQ = -(1/RC)dt

now integrate..

ln|Q| = -(t/RC)+c

exponentiate...

Q = e^(-(t/RC)+c)

but this is slightly different from the result i got with integ factor??

 

[Cyosis]

No it is the same expression, because e^c=constant.