Formulario Ecuaciones Diferenciales - UNMSM- Prof. SOTO SOTO PALERMO

Ec.dif. homogenea
f(λx,λy)=f(x.y) y'=fx,y yx=v 1f1,v-vdv-lnx=k
Ec. dif. exacta My=Nx
x0xMx,ydx+y0yN(x0,y)dy=k x0xMx,y0dx+y0yNx,ydy=k
Factor integrante
μx=eMy-NxNdx μy=eNx-MyMdx
μ(x,y)=eNx-MyMX-NY d(xy) (xαyβMx,y) y= (xαyβNx,y) x
Ec. dif. lineal y'+pxy=qx
y=e-p(x)dxep(x)dx.q(x)dx+c
Ec. dif. de Bernoulli y'+pxy=qxyn
v=y1-n v'+1-np(x)v=1-nq(x)
Ec. dif. de Clariaut y=xy'+fy' y'=dydx=p
Sol. paramtrica y=cx+fc
Sol. singular x+f'(p)=0 y=xp+f(p)
Ec. dif. de Lagrange y=xgy'+fy' y'=dydx=p
Sol. paramtrica Ѱ(x,y,c)=0
Ec. dif. de Riccati y'+a2xy2+a1xy+a0x=0
y=y1x+1z(x) z'-2a2xy1x+a1xz=a2x
APLICACIONES X=x-1y' Y=y-xy'
Lt: Y-y=y'X-x
LN: Y-y=-1y'X-x
Fam. de Trayectorias Ortogonales
Mx,y+Nx,y-dxdy=0
Ec. dif. de orden superior
anyn+an-1yn-1+…+a2y''+a1y'+a0y=fx
METODO DE VARIACION DE PARAMETROS
y(n)+pn-1yn-1+…+p2y''+p1y'+p0y=fxan=gx
Sol. de la ec. dif. homogenea
yh(x)=c1y1x+c2y2x+…+cnynx
Sol. particular de la ec. dif.
yp(x)=μ1(x)y1x+μ2(x)y2x+…+μn(x)yn(x)
Wronskiano
wy1,y2,…,yn=dety1 yn y1n-1 ynn-1
wkx=dety1…0…yn …0… y1n-1…1…ynn-1
μk(x)= xgt.wk(t)w(t)dt
E. dif. de Euler
anxnyn+an-1xn-1yn-1+…+a2x2y''+a1xy'+a0y=fx
x=et y=yt
y'=e-ty't
y''=y''t-y'te-t
y'''=y'''t-3y''t+2y'te-t
y'v=(y'vt-6y'''t+11y''t-6y'(t))e-t
METODO DE OPERADORES
anDN+an-1Dn-1+…+a2D2+a1D+a0(y)=fx
CASO1: fx=RO constante
a0=0 ypx=ROa0
a0 0 anDN+an-1Dn-1+…+ak+1Dk+1+akDk+y=fx yp(x)=ROxkakk!
CASO2:D2+a2n(y)=cos (βx)
n=1 a β ypx=1a2-β2cosβx
n>1 a=β yp(x)=xn(2a)nn!cos (αx-12nπ)
CASO3:D2+a2n(y)=sen (βx)
n=1 a β ypx=1a2-β2senβx
n>1 a=β yp(x)=xn(2a)nn!sen(αx-12nπ)

CASO 4:anDN+an-1Dn-1+…+ak+1Dk+1+akDk+y=F(D)(y)=eax
F(a) 0 ypx=1F(a)eax
F(a)=0 F(D)(y)=(D-akgD)(y)=eax g(a) 0 ypx=xkk!g(a)eax
CASO 5:F(D)(y)=eaxv(x)
ypx=1F(D)eaxv(x) ypx=eax1F(D+a)v(x)