GIA'I H^E. PHU.O.NG TRNH

201)

Gia'i h^e. phu.o.ng trnh:

dydt =x−2y+ 2 sint.HD gia’i:

Phu.o.ng trnh da.c tru.ng co cac nghi^e.m

^λ

¹

^=−1;λ

²

^{= 1}

. Tu. do du.o..c h^e.

nghi^e.m co. ba'n:

^e_e

^−t

^−t

^{; 3e}_e

^t

^t

^.(x =C

1

e

^−t

+ 3C

2

e

^t

V^a.y nghi^e.m t^o'ng quat:

y =C

₁

e

^−t

+C

₂

e

^t

.(C

⁰

₁

= 3e

^t

sint(C

⁰

₁

e

^−t

+ 3C

⁰

₂

e

^t

= 0

Bi^en thi^en hang s^o:

C

⁰

₁

e

^−t

+C

⁰

₂

e

^t

= 2 sint. ⇐⇒C

⁰

₂

=e

^−t

sint.C

₁

(t) = 3

Gia'i ra:

2e

^t

(sint−cost)C

₂

(t) =−12e

^−t

(sint+ cost).(x(t) =C

₁

e

^−t

+ 3C

₂

e

^t

−3 cost

V^a.y nghi^e.m t^o'ng quat cu'a h^e.:

y(t) =C

₁

e

^−t

+C

₂

e

^t

+ sint−2 cost.