Solutions to Applications of Higher Order Differential Equations (2) - Part B

2) For the circuit shown, consider the special case of NO capacitor. Then the differential equation of interest becomes:


L (dI/dt) + RI = E


It’s instructive to first solve this form where E is presumed constant. Begin by re-arranging the DE to get:

L dI = (E – RI) dt so that:  ò LdI/ (E – RI) = ò  dt

Let u = (E – RI) ,    Then: du = - RdI and dI = -du/ R

So:   - L/R  ò du/u = ò  dt

 

 

Now, u = E – RI and this has a (+ve) value (e.g. uo = E) at t = 0 and we assume it remains positive so çu ç = u and:

 

 

(-L/R) ln u  = t + C

 

 

Or: -(L/R) ln (E – RI)  = t + C

 

 

The constant of integration C can be found knowing that I = 0 at t = 0 hence:

 

 

C = - (L/R)  ln E

 

After suitable substitutions and some algebra:

 

L/R [ln E – ln (E – RI)] = t

 

ln [E/ E – RI]  = Rt/ L

 

Taking natural logs:

 

E/ (E – RI)  = exp (Rt/L)

 

 

Which can also be written:  (E – RI)/ E = exp (- Rt/L)

Solving for I:   I = E/R [1 - exp (-Rt/L)]

 

 

Then the current (for t > 0) always increases toward the steady state value of E/R. Now, what about the solution if we let E = E_o sin wt? (Oscillatory voltage). 

 

 

It should be fairly clear from past articles –blog posts that the solution must have the form: A sin wt + B cos wt

 

Using the integrating factor, the solution we obtain (note this is not a higher order DE!) is:

 

 

I exp (at)  =  ò exp (at)  E dt  + C

 

Then one will obtain:

 

 

 

I =   E_o { RL sin wt -  wL ² cos wt}/ (R² +  w² L ²   )  +  k exp (-Rt/L)

 

 

This can also be done using the Laplace transform technique, after re-arranging the equation to get:

 

dI/dt + (R/L) I =  E_o sin wt

 

Then write the transformed eqn. as, using  a = R/L:

 

 

s I(s) + a I(s) =   E_o  w/ (s² +  w² )

 

 

Whence:

 

 

I(s) [ s + a] =   E_o  w/ (s² +  w² )

 

 

I(s) =  E_o  w  / (s + iw) (s - iw)  (s + a)

 

For which a partial fractions decomposition can be used, followed by the inverse Laplace transform.