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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 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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