formulas Flashcards
eulers formula
x(∂u/∂x)+y(∂u/∂y)
=nu
x^2(∂^2u/∂x^2)+
2xy(∂^2u/∂x∂y)+
y^2(∂^2u/∂y^2)
=n(n-1)u
homogeneous form (x^ng(y/x))
or (y^ng(x/y))
degree of x in the following equation
𝑥^3sin^-1(y/x)
(y^2-x^2)^
1/2sin^-1(y/x)
n=3, n=1 as x^3, and x^1
chain rule with multivariable formula
∂u/∂t=
∂u/∂xdx/dt+
∂u/∂ydy/dt
(x,y,z)
∂u/∂t=
∂u/∂xdx/dt+
∂u/∂ydy/dt+
∂u/∂z*dz/dt
if (x,y)=> (r,s) two variables =
∂u/∂r=
∂u/∂xdx/dr+
∂u/∂ydy/dr
∂u/∂s=
∂u/∂xdx/ds+
∂u/∂ydy/ds
or in terms of
∂u/∂x and ∂u/∂y
∂u/∂x=
∂u/∂rdr/dx+
∂u/∂sds/dy
∂u/∂y=
∂u/∂sds/dy+
∂u/∂rdr/dy
du/dx formula
du/dx=du/dx+
du/dy*dy/dx
dy/dx=-ux/uy
Taylors and Maclaurin’s formula
f(x,y) =f(a,b) +1/1![(x-a)Fx(a,b)+(y-b)Fy(a,b)]
1/2![(x-a)^2Fxx(a,b)+(y-b)^2Fyy(a,b)+2(x-a)*(y-b)Fxy(a,b)]
1/3![(x-a)^3Fxxx(a-b)+(y-b)^3Fyyy(a-b)+3((x-a)^2(y-b)Fxxy(a,b)+3(x-a)(y-b)^2Fxyy(a,b))]…
what is Maclaurin’s theorm
maclaurens theorm is a specail case of taylors series when the expansion is about the origin(0,0)
sinhx, coshx in terms of e^x
e^x-e^-x/(2) coah
e^x+e^-x/(-2) sinh
if e^0 in Particular integral what is rhs
D->0
sinx and cosx in terms of e^ax
sinx=e^iax-e^iax/2i
cosx=e^iax-e^-iax/2
type 2 sin(ax+b) and cos(ax+b)
sin(2x)=> a=2, b=0
D^2=>-a^2
cos^2x= identity
1+cos2x/2
2^-x into e^x in terms of log
e^(log(2))x
and D value will be loga value
type 3 PI
R(x)/f(D)
x/D^2-2D (etc)
type 4 PI
e^2xcosx
e^axV
replacement =
D=> D+a
type 5 PI
R(x)/f(D)
R=xV
PI=(x-(f’(D)/f(D))V/f(D)
