Tugas Pd Pd An.docx

Mata Kuliah

: Persamaan Diferensial

Tugas Halaman 190

13. y + px = 0 Penyelesaian : y + px = 0 y = - px . . . (1)

Didifferensir terhadap x 𝑑𝑦 𝑑𝑥

𝑑𝑝

= -x 𝑑𝑥 - p 𝑑𝑝

p = -x 𝑑𝑥 - p 𝑑𝑝

2p = -x 𝑑𝑥 1

𝑑𝑝

2

𝑑𝑥

p=- x 𝑑𝑥 𝑥

1 𝑑𝑝

=-2

𝑝 1

ln 𝑥 = - 2 ln 𝑝 + ln 𝑐 1

ln 𝑥 = - ln 𝑝2 + ln 𝑐 x=

𝑐 1

𝑝2 𝑐

√𝑝 = 𝑥 𝑐2

p = 𝑥2

. . . (2)

Substitusikan (2) ke (1) y = - px 𝑐2

y = - ( 𝑥2 ) x y=-

𝑐2 𝑥

xy = - 𝑐 2 → Jawab Umum

xy = c

Jawab Singular: x=0 y=0

7.

𝑥 2 𝑝2 + 3xyp + 2𝑦 2 = 0 Penyelesaian : 𝑥 2 𝑝2 + 2xyp + xyp + 2𝑦 2 = 0 xp (xp + 2y) + y (xp + 2y) = 0 (xp + y)(xp + 2y) = 0 (xp + y) = 0

. . . (2)

(xp + 2y) = 0

. . . (3)

𝑑𝑦

Misal p = 𝑑𝑥

# Substitusikan p ke (2) xp + y = 0 𝑑𝑦

x ( 𝑑𝑥 ) + y = 0 𝑑𝑦

x 𝑑𝑥 = - y 𝑑𝑦 𝑦

=-

𝑑𝑥 𝑥

∫

𝑑𝑦 𝑦

=-∫

𝑑𝑥 𝑥

ln y = - ln x + ln c 𝑐

y=

𝑥 𝑐

y-

=0

𝑥

# Substitusikan p ke (3) xp + 2y = 0 𝑑𝑦

x ( 𝑑𝑥 ) + 2y = 0 𝑑𝑦

x 𝑑𝑥 = - 2y 𝑑𝑦

=-2

𝑦

∫

𝑑𝑦 𝑦

𝑑𝑥 𝑥

=-2∫

𝑑𝑥 𝑥

ln y = - 2 ln x + ln c y= y-

𝑐 𝑥2 𝑐 𝑥2

=0

Jadi, Jawab Umumnya yaitu : (y -

𝑐 𝑥

) (y -

𝑐 𝑥2

)=0

(xy – c) (𝑥 2 y – c ) = 0