ecuaciones

❊❥❡r❝✐❝✐♦s r❡s✉❡❧t♦s ❞❡ ❡❝✉❛❝✐♦♥❡s ❞✐❢❡r❡♥❝✐❛❧❡s
❍✉❣♦ ▲♦♠❜❛r❞♦ ❋❧♦r❡s
✶✸ ❆❜r✐❧ ✷✵✶✶

✶

❊❝✉❛❝✐♦♥❡s ❞✐❢❡r❡♥❝✐❛❧❡s ❞❡ ♣r✐♠❡r ♦r❞❡♥

✶✳✶

❊❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s ② r❡❞✉❝✐❜❧❡s ❛ ❡st❛s✳

✶✳
dy
+ 2y = 0
dx
❉❡❢✐♥✐♠♦s ❡❧ ❢❛❝t♦r ✐♥t❡❣r❛♥t❡✳

p(x) = 2

´

❢❛❝t♦r ✐♥t❡❣r❛♥t❡✿ e 2dx ❂ e2x
♠✉❧t✐♣❧✐❝❛♠♦s ❧❛ ❡❝✉❛❝✐♦♥ ♣♦r ❡❧ ❢❛❝t♦r ✐♥t❡❣r❛♥t❡✳
dy
e2x dx
+ 2e2x = 0

❡❧ ❧❛❞♦ ✐③q✉✐❡r❞♦ ❞❡ ❧❛❡❝✉❛❝✐♦♥ s❡ r❡❞✉❝❡ ❛✿
d
2x
dx [e y]

=0

s❡♣❛r❛♠♦s ✈❛r✐❛❜❧❡s ❡ ✐♥t❡❣r❛♠♦s✳
´

d
2x
dx [e y]

=0

´

dx + c

e2x y = c
y = ce−2x

✷✳

dy
= 3y
dx

❢♦r♠❛ ❧✐♥❡❛❧✳
dy
dx

− 3y = 0

p(x) = −3
´

❋❛❝t♦r ✐♥t❡❣r❛♥t❡✿ e −3dx ❂e−3x
♠✉❧t✐♣❧✐❝❛♠♦s ♣♦r ❢❛❝t♦r ✐♥t❡❣r❛♥t❡✳
✶

dy
e−3x dx
− 3e−3x y = 0

´

´
= 0 dx + c

dy −3x
y
dx [e

e−3x y = c
y = ce3x

✸✳3

dy
+ 12y = 4
dx

♣❛s❛♠♦s ❧❛ ❡❝✉❛❝✐♦♥ ❛ ❧❛ ❢♦r♠❛ ❧✐♥❡❛❧✳
dy
dx

+ 4y =

4
3

p(x) = 4

❋❛❝t♦r ✐♥t❡❣r❛♥t❡✿ e

´

4dx

❂e4x

dy
+ 4e4x y = 43 e4x
e4x dx
´ d 4x
´ 4x
e dx + c
dx [e y] =

e4x y = 41 e4x + c
y=

✹✳

1
4

+ ce−4x

y = 2y + x2 + 5

❢♦r♠❛ ❧✐♥❡❛❧
y − 2y = x2 + 5

❋❛❝t♦r ✐♥t❡❣r❛♥t❡✿ e

´

−2dx

= e−2x

e−2x y − 2e−2x y = e−2x x2 +5e−2x
´ d −2x
´
´
y] = e−2x x2 + 5 e−2x + c
dx [e
e−2x y = − 25 e−2x − 14 e−2x (2x2 + 2x + 1) + C
2

y = − x2 −

✺✳

x
2

−

1
4

+

5
2

+ ce2x

ydx − 4(x + y 6 )dy = 0
ydx = 4(x + y 6 )dy
dx
dy

=

4(x+y 6 )
y

❀

dx
dy

✷

=

4x
y

+

4y 6
y

❞❡✜♥✐♠♦s ❧❛ ❢♦r♠❛ ❧✐♥❡❛❧✳
dx
dy

❋❛❝t♦r ✐♥t❡❣r❛♥t❡✿ e−4

´

1
y dy

−

4x
y

= 4y5

❀ e−4 log(y) ❀ elog(y) ❀ y −4 =
−4

1 dx
y 4 dy

−

1 4x
y4 y

d 1
dy [ y4 x]

´

d 1
dy [ y4 x]
1
y4 x

1
5
y 4 4y

=

= 4y

´
= 4 ydy

= 2y 2 + C

x = 2y 6 + cy 4

✻✳

xy + y = ex
y + x1 y =

ex
x

❋❛❝t♦r ✐♥t❡❣r❛♥t❡✿
´

1
x dx

e

= elog x = x

xy + xx y =
d
dx [xy]

■♥t❡❣r❛♠♦s✿

´

d
dx [xy]

=

xex
x

= ex

´

ex dx+ c

xy = ex + c
y = ex x−1 + cx−1

✼✳
x
dy
dx

dy
2
+y = 2
dx
y

+

y
x

=

2
xy 2

✳✳✳✭✶✮

❤❛❝❡♠♦s ❧❛ s✉st✐t✉❝✐♦♥✿ u = y 1−n ❞♦♥❞❡ n = −2
u = y 1−(−2) = y 3 ❀u1/3 = y

❉❡r✐✈❛♠♦s ❡st❛ ✉❧t✐♠❛✳
1 −2/3 du
3u
dx

✸

=

dy
dx

1
y4

❙✉st✐t✉✐♠♦s ❡♥ ❧❛ ❡❝✉❛❝✐♦♥ ❞✐❢❡r❡♥❝✐❛❧ ✶✳
1 −2/3 du
3u
dx

+

u1/3
x

=

2(u1/3 )2
x

❆❝♦♠♦❞❛♠♦s ❛ ❧❛ ❢♦r♠❛❧✐♥❡❛❧✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ t♦❞❛ ❧❛ ❡❝✉❛❝✐♦♥ ♣♦r 13 u2/3 ✳
du
dx

+ 3 ux =

6
x

❊st❛ ❡s ✉♥❛ ❡❝✉❛❝✐♦♥ ❧✐♥❡❛❧✳ ❉❡✜♥✐♠♦s ❡❧ ❢❛❝t♦r ✐♥t❡❣r❛♥t❡✳
e3

´

1
x dx

3

= e3 log x = elog x = x3

▼✉❧t✐♣❧✐❝❛♠♦s ♣♦r ❢❛❝t♦r ✐♥t❡❣r❛♥t❡✳
3u
36
x3 du
dx + 3x x = x x
d
3
dx [x u]

✐♥t❡❣r❛♠♦s✳

´

d
3
dx [x u]

= 6x2

´
= 6 x2 + c

x3 u = 2x3 + c
u = 2 + cx−3

❙✉st✐t✉✐♠♦s u = y3
y 3 = 2 + cx−3
dy
+ y 3/2 = 1❀ ❝♦♥❞✐❝✐♦♥ y(0) = 4
✽✳ y 1/2 dx
dy
dx

+

y 3/2
y 1/2

=

dy
1
↔ dx
y 1/2

+ y = y −1/2

u = y 1−n ❀ n = −1/2❀ u = y 1−(−1/2) = y 3/2
u2/3 = y
2 −1/3 du
3u
dx

=

dy
dx

❙✉st✐t✉✐♠♦s✳
2 −1/3 du
3u
dx

+ u2/3 = (u2/3 )−1/2

▼✉❧t✐♣❧✐❝❛♠♦s ❧❛ ❡❝✉❛❝✐♦♥ ♣♦r 23 u1/3
du
dx

+ 23 u =

▲❛ ❡❝✉❛❝✐♦♥ s❡ r❡❞✉❥♦
´❛ ✉♥❛ ❧✐♥❡❛❧✳
3
3❋❛❝t♦r ✐♥t❡❣r❛♥t❡✿ e 2 dx = e 2 x
✹

3
2

3

3

3

3
3
2x u = e2x
e 2 x du
dx + e
2
2
3
d
2 x u]
dx [e

´

3
d
2 x u]
dx [e

=

3

= 32 e 2 x

´

3

3 32 x
dx
2e

+c

3

e 2 xu = e 2 x + c
3

u = 1 + ce− 2 x

❙✉st✐t✉✐♠♦s u = y 3/2
3

y 3/2 = 1 + ce− 2 x

✘ ❙♦❧✉❝✐♦♥ ❣❡♥❡r❛❧✳

❆❤♦r❛ ❛♣❧✐❝❛♠♦s ❧❛s ❝♦♥❞✐❝✐♦♥❡s ✐♥✐❝✐❛❧❡s✳ y(0) = 4
3

43/2 =1 + ce− 2 0
8−1=c
c=7

❙✉st✉t✉✐♠♦s ❡❧ ✈❛❧♦r ❞❡ ❝ ❡♥ ❧❛ ❡❝✉❛❝✐♦♥ ❣❡♥❡r❛❧✳
3

y 3/2 = 1 + 7e− 2 x

✾✳
y +

✘ ❙♦❧✉❝✐♦♥ ♣❛rt✐❝✉❧❛r✳

2
y = −2xy 2
x

u = y 1−n ❀ ❞♦♥❞❡ n = 2
❡♥t♦♥❝❡s✿
u = y 1−2 ❀ u = y −1 ❀ u−1 = y
−u−2 du
dx =

dy
dx

s✉st✐t✉✐♠♦s ❡♥ ❧❛ ❡❝✉❛❝✐♦♥✳
2 −1
−u−2 du
= −2x(u−1 )2
dx + x u

♠✉❧t✐♣❧✐❝❛♠♦s ♣♦r −u2
du
dx

− x2 u = 2x

❡st❛ ❡s ✉♥❛...