GIA'I PHU.O.NG TRNH

101)

Gia'i phu.o.ng trnh:

^{y” +y}

⁰

^=x+e

^−x

HD gia’i:

Phu.o.ng trnh da.c tru.ng

^λ

²

^{+λ= 0 ⇔λ}

¹

^{= 0;λ}

²

⁼⁻¹

Nghi^e.m t^o'ng quat cu'a phu.o.ng trnh thu^an nh^at:

^y=C

¹

^+C

²

^e

^−x

Tm nghi^e.m ri^eng du.o.i da.ng

^{y = y}

¹

^+y

²

, trong do

^y

¹

^{, y}

²

la cac nghi^e.m tu.o.ng u.ng

cu'a cac phu.o.ng trnh:

^{y” +y}

⁰

^=x

va

^{y” +y}

⁰

^=e

^−x

•

V

^λ

1

= 0

la nghi^e.m cu'a phu.o.ng trnh da.c tru.ng n^en

^y

¹

^=x(Ax+B)

Bang phu.o.ng phap h^e. s^o b^at di.nh du.o..c:

^y

¹

^{= 1}₂^x

²

^−x• λ

2

=−1

la nghi^e.m cu'a phu.o.ng trnh da.c tru.ng n^en:

^y

²

^=Axe

^−x

Thay vao va dung h^e. s^o b^at di.nh suy ra:

^y

²

^=−xe

^−x

Cu^oi cung NTQ:

^y=C

1

+C

₂

e

^−x

+12x

²

−x−xe

^−x