differential equations

Question

A 20-volt electromotive force is applied to an LR-series circuit in which the inductance is 0.1 henry and the resistance is 30 ohms.

L di/dt + Ri= E(t)

 Determine the values of L, R, and E(t)

 for the given situation.

L=R=E(t)

=

Find the current i(t)

 if i(0) = 0.

i(t) = 

 

Determine the current as t → ∞.

lim t→∞ i(t) = 

Answer 1

Kirchhoff’s second law states that the sum of the voltage  drop across the inductor L(di/dt) plus the voltage drop across the resistor (Ri) is the same as the impressed voltage on the circuit E(t)

L(di​/dt) + Ri = E(t)

Given,

L = 0.1H

R = 50Ω

E = 20V

So, 0.1(di/dt) + 50i = 20

divide through by 0.1

So, (di/dt) +500i = 200 …………………Equation 1

since the above is a linear differential Equation, then The integrating factor is 

= e^∫500dt = e^(500t)

Multipying equation 1 by the integrating factor

∴i x e^(500t) = ∫200 x e^(500t) dt

By integrating Right hand side

i x e^(500t) = 2​/5 e^(500t) + c

Making i(t) the subject, Divide through by e^(500t)

∴i(t) = 2/5 + (c e^(−500t) )

Applying the initial condition 

i(0)=0

then, 0 = 2/5 + c

 e0→c = −2/5

Substitute in i(t)

∴ The current i(t) if i(0) = 0  is i(t) = 2/5 −2/5e−500t   

∴limt→∞​i(t) = limt→∞​(2/5 − 2/5e^(−500t))= 2/5 − 2/5(0) = 2/5