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 𝑖`e^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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