formulas

Question 1

eulers formula

Answer 1

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))

Question 2

degree of x in the following equation
𝑥^3sin^-1(y/x)

(y^2-x^2)^1/2
1/2sin^-1(y/x)

Answer 2

n=3, n=1 as x^3, and x^1

Question 3

chain rule with multivariable formula

Answer 3

∂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

Question 4

du/dx formula

Answer 4

0=du/dx+
du/dy*dy/dx

dy/dx=-ux/uy

Question 5

Taylors and Maclaurin’s formula

what is Maclaurin’s theorm

Answer 5

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))]…

maclaurens theorm is a specail case of taylors series when the expansion is about the origin(0,0)

Question 6

sinhx, coshx in terms of e^x

Answer 6

e^x-e^-x/(2) coah
e^x+e^-x/(-2) sinh

Question 7

if e^0 in Particular integral what is rhs

Answer 7

D->0

Question 8

sinx and cosx in terms of e^ax

cos^2x= identity

2^x into e^x in terms of log

Answer 8

sinx=e^iax-e^iax/2i
cosx=e^iax-e^-iax/2

cos^2x=> 1+cos2x/2

e^(log(2))x

and D value will be loga value

Question 9

2^-x into e^x in terms of log

Answer 9

e^(log(2))x

and D value will be loga value

Question 10

type 2

type 3 PI

type 4 PI

type 5 PI

Answer 10

TYPE 2
sin(ax+b) and cos(ax+b)

sin(2x)=> a=2, b=0
D^2=>-a^2
TYPE 3
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)

Question 11

method of variation of parameters ISA question

Answer 11

y’‘+py’+qy =R(x)
they will ask what is P and Q in ISA

A=-∫Y2R(X)dx/w
B=-∫Y1R(X)dx/w
w is matrix

[Y2 Y1][Y2’Y1’] (top and bottom matrix) => w
w=wronksions scale
final answer of both intregation and A, B should be in the for (A)Y1+(B)Y2