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# In a Circuit Containing Inductance L, Resistance R, and Voltage E, the Current I is Given by L D I D T + R I = E .Find the Current I at Time T at T = 0 and I = 0 and L, R and E Are Constants. - Applied Mathematics 2

In a circuit containing inductance L, resistance R, and voltage E, the current i is given by L (di)/dt+Ri=E.Find the current i at time t at t = 0 and i = 0 and L, R and E are constants.

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#### Solution

The given equation (di)/dt+(Ri)/L=E/L  is linear of the type dy/dx+py=Q

∴ Its solution is 𝑖e^int"^R/L^dt=int e^int^"R/L^dt. E/L. dt+c

i.e^Rt/L= E/L int e^(rt)/L dt+C=. E/L.e^(rt)L L/R+C

= E/R e^(Rt)/L+c

When t = 0 and i=0 ∴ C= -E/R

∴ i.e^(Rt)/L=E/R e (Rt)/L-E/R

∴ i= E/R (e^(Rt)/(L-1))

∴ i=E/R (1-e^(-Rt)/L)

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
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