- - -
: y'=2y-4z, z'=y-3z+e^x >
> >
Lutik
, :
y'=2y-4z
z'=y-3z+e^x

2- :
-=-z'-3z+e^x
y=z'+3z-e^x
1- :
y'=z''+3z'-e^x

z''+3z'-e^x=2y-4z
z''+3z'+4z=2y-e^x

,

z=e^kx
z'=k*(e^(kx))
z''=k^2*(e^(kx))

k^2*(e^(kx))+3*k*(e^(kx))+4*e^(kx)=0
e^(kx)(k^2+3k+4)=0
k^2+3k+4=0
k1=(-3+7i)/2
k2=(-3-7i)/2
, z1=e^(-3x)cos(7x), z2=e^(-3x)sin(7x)
z=1*e^(-3x)cos(7x)+2*e^(-3x)sin(7x)

z' z'=y-3z+e^x ?

-
.
:
z''+3z'+4z=2y-e^x
z''+3z'+4z=2y+e^x
y, :
y=z'+3z-e^x,
, x.
Lutik
y=z'+3z-e^x

y=-3*e^(-3x)*c1*cos(7x)-7*c1*e^(-3x)*sin(7x)-3*c2*e^(-3x)*sin(7x)+7*c2*e^(-3x)*cos(7x)+3*c1*e^(-3x)cos(7x)+3*c2*e^(-3x)sin(7x)-e^x

y=-7*c1*e^(-3x)*sin(7x)+7*c2*e^(-3x)*cos(7x)+3*c1*e^(-3x)cos(7x)+-e^x

z y
y=-7*c1*e^(-3x)*sin(7x)+7*c2*e^(-3x)*cos(7x)+3*c1*e^(-3x)cos(7x)+-e^x

z=1*e^(-3x)cos(7x)+2*e^(-3x)sin(7x)
-
. , ,
z''+3z'+4z=2y+e^x
y, z,z' x, ..
z''+3z'+4z=2(z'+3z-e^x)+e^x.
Dimka
smile.gif
.
Lutik
z''+3z'+4z=2(z'+3z-e^x)+e^x
z''+3z'+4z-2z'-3z=0
z''+z'+z=0

z=e^kx
z'=k*e^kx
z''=(k^2)*e^(kx)

(k^2)*e^(kx)+k*e^kx+e^kx=0
e^(kx)*(k^2+k+1)=0
k1=(-1+15i)/2
k2=(-1-15i)/2
:
z1=e^(-x)cos(15x)
z2=e^(-x)sin(15x)
z=e^(-x)cos(15x)+e^(-x)sin(15x)
Dimka
Lutik
, 4*1 4*4, smile.gif

z''+3z'+4z=2(z'+3z-e^x)+e^x
z''+3z'+4z-2z'-3z=0
z''+z'+z=0

z=e^kx
z'=k*e^kx
z''=(k^2)*e^(kx)

(k^2)*e^(kx)+k*e^kx+e^kx=0
e^(kx)*(k^2+k+1)=0
k1=(-1+5i)/2
k2=(-1-5i)/2
:
z1=e^(-x)cos(5x)
z2=e^(-x)sin(5x)
z=e^(-x)cos(5x)+e^(-x)sin(5x)
Dimka
.
Lutik


z''+3z'+4z=2(z'+3z-e^x)+e^x
z''+3z'+4z-2z'-3z=0
z''+z'+z=0

z=e^kx
z'=k*e^kx
z''=(k^2)*e^(kx)

(k^2)*e^(kx)+k*e^kx+e^kx=0
e^(kx)*(k^2+k+1)=0
k1=(-1+3i)/2
k2=(-1-3i)/2
:
z1=e^(-x)cos(3x)
z2=e^(-x)sin(3x)
z=e^(-x)cos(3x)+e^(-x)sin(3x)
Dimka
z=e^(-x)(1cos(3x)+2sin(3x))

y

.
Lutik
z': y=z'+3z-e^x
z'=-e^(-x)*1*cos(3x)-3*e^(-x)*1sin(3x)-e^(-x)*2*sin(3x)+3*e^(-x)*1*cos(3x)

y=-e^(-x)*1*cos(3x)-3*e^(-x)*1*sin(3x)-e^(-x)*2*sin(3x)+3*e^(-x)*2*cos(3x)+3*e^(-x)*1*cos(3x)+3*e^(-x)*2*sin(3x)-e^x

y=2*e^(-x)*1*cos(3x)-3*e^(-x)*1*sin(3x)+2*e^(-x)*2*sin(3x)+3*e^(-x)*2*cos(3x)-e^x
Dimka
(Lutik @ 4.12.2009, 23:47) *



z''+3z'+4z=2(z'+3z-e^x)+e^x
z''+3z'+4z-2z'-3z=0
z''+z'+z=0


. . .
Lutik
-

z''+3z'+4z=2(z'+3z-e^x)+e^x
z''+3z'+4z-2z'-6z=-e^x
z''+z'-2z=-e^x

z=e^kx
z'=k*e^kx
z''=(k^2)*e^(kx)

(k^2)*e^(kx)+k*e^(kx)-2*e^(kx)=0
e^(kx)*(k^2+k-2)=0
k^2+k-2=0
D=9
k1=(-1+3)/2=1
k2=(-1-3)/2=-2

z1=e^0x=1
z2=e^-2x

z=c1+c2*e^-2x
-
(Lutik @ 5.12.2009, 0:28) *

k1=(-1+3)/2=1
k2=(-1-3)/2=-2

z1=e^0x=1
z2=e^-2x

?
Lutik
,
k1=(-1+3)/2=1
k2=(-1-3)/2=-2

z1=e^x
z2=e^-2
z=c1*e^x+c2*e^-2x

y=z'+3z-e^x z' y, , . z y
-
, , .
Lutik
, , -e^x
z''+z'-2z=-e^x
z*=(Ax^2+Bx)*e^x,

z*'=(e^x)*A*(x^2)+2*(e^x)*Ax+Bx*e^x+B*e^x
z*''=(e^x)*A*(x^2)+4*(e^x)*A*x+2*(e^x)*A+(e^x)*B*x+2*(e^x)*B

(e^x)*A*(x^2)+4*(e^x)*A*x+2*(e^x)*A+(e^x)*B*x+2*(e^x)*B+(e^x)*A*(x^2)+2*(e^x)*Ax
+Bx*e^x+B*e^x-2*(A*(x^2)*e^x+B*x*e^x)=-e^x

6*(e^x)*A*x+2*(e^x)*A+3*(e^x)*B=-e^x
6A=-1
A=-1/6
=-2/9
, , 6=-1,

:
6A=0 => A=0
3*B=-1
b=-1/3

z*=(-1/3)*x*e^x
-
(Lutik @ 5.12.2009, 14:46) *

z*=(Ax^2+Bx)*e^x, ...

?
Lutik
-e^x f(x)=P(x)e^Lx, L k1=1, (Ax^2+Bx)*e^x,
Lutik
-e^x f(x)=P(x)e^Lx, L k1=1, (Ax^2+Bx)*e^x,
-
k=1 - ,, P(x)*e^(kx), z=x*Q(x)*e^(kx).
Lutik
Q(x) -e^x
-
P(x)=-1 - , Q(x) - ,.. Q(x)=C.
Lutik
z=x*A*e^x
z'=A*x*e^x+A*e^x
z''=2*A*e^x+x*A*e^x

z''+z'-2z=-e^x
2*A*e^x+x*A*e^x+A*x*e^x+A*e^x-2*x*A*e^x=-e^x
3*A*e^x=-e^x
A=-1/3

z=(-1/3)*x*e^x -
-
.
Lutik
y=z'+3z-e^x ,
z'=(-1/3)*x*(e^x)-(1/3)*(e^x)

y=(-1/3)*x*(e^x)-(1/3)*(e^x)+3*(-1/3)*x*(e^x)-(e^x)
y=(-4/3)*x*(e^x)-(4/3)*(e^x)
.
!!!
wils
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wils
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