GIA'I PHU.O.NG TRNH (2X+ 1)Y” + (2X−1)Y0−2Y=X2+X BI^ET RANGNO CO...

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Gia'i phu.o.ng trnh

^(2x+ 1)y” + (2x−1)y

⁰

−2y=x

²

+x

bi^et rang

no co hai nghi^e.m ri^eng

^y

¹

^{= x}

²

^{+ 4x}₂ ^{−1; y}

²

^{= x}

²

₂^{+ 1}

.

HD gia’i:

Tu. hai nghi^e.m ri^eng

^y

¹

^{, y}

²

cu'a phu.o.ng trnh ta suy ra nghi^e.m ri^eng cu'a

phu.o.ng trnh thu^an nh^at la

^y

1

=y

₁

−y

₂

= 2x−1

Suy ra nghi^e.m thu. hai:

Z 1y

₂

=y

₁

y

₁

²

e

⁻

^R

^p(x)dx

dx= (2x−1)(2x−1)

²

e

⁻

^R

^2x−1

^2x+1

^dx

dxZ (2x+ 1)e

^−x

Z e

^−x

(1−2x)= 2(x−1)(2x−1)

²

dx= 12(2x−1)[−(2x+ 1)e

^−x

(2x−1)

²

+2x−1 dx]=−e

^−x

Suy ra NTQ:

^{y =C}

1

(2x−1) +C

₂

e

^−x

Va nghi^e.m t^o'ng quat cu'a phu.o.ng trnh ban d^au:

^y=C

¹

^{(2x−1) +C}

²

^e

^−x

^{+ x}

²

₂^{+ 1}