Review useful concepts for DEs 1
1 Concepts
1.1 Trigonometric functions
sin2x+ cos2x= 1
sin(x+y) = sin xcos y+ cos xsin y
sin(2x) = 2 sin xcos x
cos(2x) = 2 cos2x−1
1.2 Exponential function and natural
log
eln x=x; ln ex=x
ex+y=exey, ex/ey=ex−y,(ex)p=exp
1.3 Derivative of elementary func-
tions
I) power of x
d
dxxp=pxp−1
II) trigonometric functions
d
dx sin x= cos x, d
dx cos x=−sin x.
III) exponential and natural logrithm
d
dxex=ex,d
dx ln x=1
x.
IV) chain rule
d
dxf(g(x)) = f0(g(x))g0(x)
1.4 Integrals
I)
Z1
xdx = ln |x|+c
II) Using chain rule in integral
Zf0(g(x))g0(x)dx =Zf0(y)dy
=f(y) + c=f(g(x)) + c
where we note y=g(x).
III) integration by parts
We know
(fg)0=f0g+fg0→f0g= (fg)0−f g0
or
(fg)0=f0g+fg0→(f0dx)g=d(fg)−f(g0dx)
So
Zf0(x)g(x)dx =f(x)g(x)−Zf(x)g0(x)dx
1.5 Complex number
I) i=√−1, i2=−1.
II) z=x+yi,x= Re(z), y= Im(z).
III) complex conjugate of zis ¯z, ¯z=x−yi.
IV) |z|=px2+y2= (z¯z)1
2.
V) Euler’s formula
eiθ = cos θ+isin θ
VI) Partial fraction
3
(s−2)(s+ 1) =A
s−2+B
s+ 1
What are Aand B?
1.6 Curves in space
I) How can we parametrize a curve in the plane?
II) Can you visualize (x(t), y(t)), t∈[a, b] if x(t)
and y(t) are given?
III) What is level curves of a function?
IV) What does it mean by saying we take derivative
about a function along a path?
