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Apuntes ecuaciones diferenciales segundo orden
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Apuntes ecuaciones diferenciales segundo orden
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Ecuaciones diferenciales de segundo orden yorden superior
General homogénea
y'
"
+
1) 3
y
=
D D =
32 D 0-(D 1) 0
+
+
=
D
=
=
- 1
*x
+
y C1.e c2.
=
+
2)
y y" =0
D =
- 1(m 1) =
13
+ =
D 1 - D(D 1) 0
+
=
D 0 =
cm 2) =
1X
e Cs.e
-
*
C1.e C2-X-
y
+ +
=
+
3) y" y 3
=
+1 3 i
B = -
D -
=
1 0
= =
y C1.ex
=
·COS 1X+ cz.esen 1X
e -
parte parte
real imaginaria
Variación de constantes. Wronskiano
+
y" y
=x
D D =
12 +
B 0 =
-
D(D 1) 0 +
=
D -
= 1
1X
Y GH C1-e
=
(2. -
+
1X
e
x -
yp v
= .
Vz.e
+
Ve Ve
1X -
A I ultima
-
e colocamos o hasta la se CDIOCA
gue
el resultado que te dan de la ecuación.
0 -
efx X
-
derivamos
xe*, dx
* =
=X
e V
ex
V
f x.e
-
=
-
- =
=
, fx
+
dx
*x
V ( *) 0-v
-
-
+ -
x- e =
x- e
=
v =
-
e
((x- e dx) e +()
+ + *
+
ax).e
7x
+
yo c.e (2.e xe-
-
- -
-
=
Ejemplo:
4y" 36y +
COSeC
=
3t 1 =
sen 3t
y" +3,y 1cosec
=
St =
senst
D2 1 0
+
=
c B = -
1 -D 1 =
-
1 0
=
i
=
y C1.e.COS 1x c2.e.sen 1X
+
=
e.COS 1x V2 e-sen 1X
y v. +
.
=
COS 1X sen 1X Ve D
Esenst
=
-
sen COS
1x X vi
COS 1X. VI'+sen 1XV2' 1
=
1X.VI'+cos X. Ve 1 1
sen
-
=
4 sen 3t
Vi =
-sen 1X Ve' =
-
tg (COS X) Ve
CDS 1X
sen x + cosx) v2+COS Ve
X =
sestsVe' (1+sen x tg(Osx))
1
u
-
sen 3t
V2 =
I 1
4 sen3x 1+ se+gcos x)
dX
Vi 1
-
=
tg(CDS X)
4 sen 3t ( 3
v=
1
=
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