ListaCalcII 1

❊①❡r❝í❝✐♦s ❞❡ ❈á❧❝✉❧♦ ■■
✶

dy
= y log x❀
✭♥✮ x dx

❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s

✭♦✮

♦r❞✐♥ár✐❛s

dy dx =

x+y x−y ✳

❘❡s♣✿ 2 tan−1 ( xy ) = log(x2 + y 2 ) +
C❀

✶✳✶ ❙❡♣❛rá✈❡✐s ❡ ❤♦♠♦❣é✲
♥❡❛s

✭♣✮

dy dx =

xy x2 +2y 2

✶✳ ❘❡s♦❧✈❛ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❛❜❛✐①♦✳

✭q✮

dy dx =

❘❡s♣✿

x2 +xy+y 2 x2 tan−1 ( xy )

dy dx =

x3 +3xy 2
3x2 y+y 3

✭❛✮

dy dx =

y
2x

❀

✭r✮

❘❡s♣✿ y 2 = Cx❀ dy ✭❜✮ dx
= 3y−1
❀
x
✭❝✮

dy dx =

x2 y2 dy dx =

✳

❘❡s♣✿
✭❢✮
✭❣✮

dy dx dy dx ✭t✮

+C

=x y ❀
= 2y ✳

dy dx = 1 − y2✳

❘❡s♣✿ y =
✭❥✮
✭❦✮
✭❧✮
✭♠✮

= 1 + y2❀

dy dx dy dx = y 2 (1 − y)❀

dy dx =

y x − e− x ✳ y ❘❡s♣✿ y 3 + 3y − 3x2 = 24✳

Ce2x −1
Ce2x +1

dy dx dy dx ❀

✸✳ ❆❝❤❡ ✉♠❛ ❡q✉❛çã♦ ❞❛ ❝✉r✈❛ ❞♦ ♣❧❛♥♦ xy q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (2, 3) ❡ t❡♠✱
❡♠ ❝❛❞❛ ♣♦♥t♦ (x, y)✱ ✐♥❝❧✐♥❛çã♦ ✐❣✉❛❧ ❛
2x
✳
1+y 2

x2

❘❡s♣✿ y = Ce 2 ❀ dy ✭❤✮ dx
= ey sin x❀
✭✐✮

= log |x| + C ❀

✷✳ ▼♦str❡ q✉❡ ❛ ❝✉r✈❛ x2 − y 2 = c✱ ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ❞❡ c✱ s❛t✐❢❛③ ❛ ❡q✉❛çã♦
❞✐❢❡r❡♥❝✐❛❧ dx
= xy ❡♠ t♦❞♦s ♦s s❡✉s ♣♦♥✲ dy t♦s ✭♥♦t❡ q✉❡ ❛ ❝✉r✈❛ é ✉♠❛ ❝✉r✈❛ ❞❡
♥í✈❡❧✮✳

✳

x2 y3 3 y4 = x3
4
2 3

✳

dy
✭s✮ x dx
= y + x cos2 ( xy )✳
❘❡s♣✿ y = x tan−1 (log |Cx|)❀

❘❡s♣✿ x3 − y 3 = C ❀ dy = x2 y 2 ❀
✭❞✮ dx
✭❡✮

❀

✹✳ ❘❡♣✐t❛ ♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r ♣❛r❛ ♦ ♣♦♥t♦
✳
(1, 3) ❡ ✐♥❝❧✐♥❛çã♦ 1 + 2y x =2+e ✳
❘❡s♣✿ y = − log(Ce−2x − 12 )❀ y ✺✳ ▼♦str❡ q✉❡ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ξ = x−x0 ❡ η = y −y0 tr❛♥s❢♦r♠❛ ❛ ❡q✉❛çã♦

= sin x cos2 y ✳
❘❡s♣✿ y = tan−1 (C − cos x) + nπ ❀

dy ax + by + c
=
dx ex + f y + g

✶

✶

✷

❊◗❯❆➬Õ❊❙ ❉■❋❊❘❊◆❈■❆■❙ ❖❘❉■◆➪❘■❆❙

♥❛ ❡q✉❛çã♦ ❤♦♠♦❣é♥❡❛

✭❝✮

dη aξ + bη
=
dξ eξ + f η

dy dx + (cos x)y = 2xe− sin x y(π) = 0

❘❡s♣✿ y = (x2 − π 2 )e− sin x ❀
1

s❛❜❡♥❞♦ q✉❡ (x0 , y0 ) é ❛ s♦❧✉çã♦ ❞♦ s✐s✲ t❡♠❛ ax + by + c = 0 ex + f y + g = 0.

✻✳ ❯s❡ ❛ té❝♥✐❝❛ ❞♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r ♣❛r❛ dy = x+2y−4
✳
r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ dx
2x−y−3

✶✳✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s
❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠
✼✳ ❘❡s♦❧✈❛ ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s ❞✐❢❡r❡♥✲
❝✐❛✐s✿
✭❛✮

= x2 ✳
❘❡s♣✿ y = x3 + cx2 ❀

✭❜✮

dy dx ✭❝✮

− 2y = 3✳
❘❡s♣✿ y = 32 + Ce−2x ❀

dy dx −