Ecuaciones Diferencialess Circuitos

CIRCUITOS ELÉCTRICOS
R=〖10〗^6 Ω ,L=1H y E=1 V ,con i(0)=0 .Halle la corriente i en cualquier instante t.
Ldi/dt+R i=E(t)
di/dt+R/L i=1/L E(t)
〖µ(x)=e〗^∫▒〖R/L dt〗=e^((R/L)t)
e^((R/L)t) di/dt+e^((R/L)t) R/Li=e^((R/L)t) 1/L E(t)
∫▒〖e^((R/L)t) t= ∫▒〖e^(R/L)t 1/L E(t〗)〗
i= e^((-R/L)t)/L ∫▒〖e^(R/L)t 1/LE(t〗)+Ce^((-R/L)t)
E(t)=1
i= e^((-R/L)t)/L ∫▒〖e^(R/L)t dt〗 +Ce^((-R/L)t)
i= 1/R+C e^((-R/L)t)
0=1/R+C e^((-R/L)t)
C=-1/R
i= 1/R-1/R e^((-R/L)t)
i(t)= 〖10〗^(-6)-〖10〗^(-6) e^(-〖10〗^(-6) t)







Determine la corrienteI (t) de un circuito cuando L = .005 H , R= 1W y C=.02 faradios.


E(t)=100 [1-u (t-1)] ; I(0)=0

Ldi/dt+R i 1/C ∫_0^t▒〖i (t)dt=100〗 [1-u (t-1)]

(.05)di/dt+R i 1/(.02) ∫_0^t▒〖i (t)dt=100〗 [1-u (t-1)]

.05(S i,-i(0))+ i,+ 50i/S=(1/S- e^(-s)/S)

SI+200 i ,+ (10000 i)/S=20000(1/S- e^(-s)/S)i((S^2+200S+10000)/S)=20000 (1/S- e^(-s)/S)

i=20000S/(S+100)^2 ((1-e^(-S))/S)= 20000/(S+100)^2 -(20000e^(-S))/(S+100)^2

i (t)=20000 t e^(-100t)- 20000 t e^(-100t) e^(-S)

i ( t)=20000 t e^(-100t)- 20000 (t-1) e^(-100t) u(t-1)