Preparation problems for Math Method in Physics (3) course
Dr. Ameenah Al-Ahmadi 1
I. Ordinary differential equations First-order differential equations
Separable variables 1) Solve the differential equation
y/x y'
2) Solve Laplace’s equation ∇2V = 0 using separation: V (x, y) = Vx(x) Vy(y) 3) Solve the initial-value problem
3
4
,y( ) y
x dx
dy
Second-order equations with constant coefficients 4) Solve the following differential equations:
a.
2
25 3 0
2
y
dx dy dx
y d
b. 2 2
0
2
k y dx
y
d
; k is a constantc. 2
6 5 3
2
y
dx dy dx
y d
5) Expand
0 0
0 f x x
x x
in a Fourier series.
II. Useful integrals you need 6) Evaluate the following integrations:
a.
x cos x dx
b.
x cos ax dx
; where a is a constant.c.
x
2cos ax dx
; where a is a constant.d.
x sin x dx
e.
x sin ax dx
; where a is a constant.f.
x
2sin ax dx
; where a is a constant.g.
sin
2ax dx
; where a is a constant.Preparation problems for Math Method in Physics (3) course
Dr. Ameenah Al-Ahmadi 2
h.
cos
2ax dx
; where a is a constant.i.
cos ax . sin bx dx
; where a & b are a constant.j.
e
axdx
; where a is a constant.k.
xe
axdx
; where a is a constant.l.
xe
ax2dx
; where a is a constant.m.
x
2e
ax2dx
; where a is a constant.III. Dirac delta function
7) Define the Dirac delta function and draw each functions
a. 𝛿(𝑥 ) =………
b. 𝛿(𝑥 − 𝑥0) =……….
8) Complete Properties of Deric delta function a.
(
0)
...
x
x
b.
( )
...
x
c.
) ( ) ...
( x x
0f x dx
d. 𝛿(𝑥) = 𝛿(– 𝑥), 𝛿(𝑥) behaves as if it were an even function (true, false) e. 𝛿(𝑥 – 𝑥0) = 𝛿(– (𝑥 – 𝑥0)), 𝛿(𝑥 – 𝑥0) is symmetric about x = x0 (true, false) f.
FT ( x ) ... ..., FT is Fourier transform
g.
FT ( x x
0) ... ...
, FT is Fourier transform IV. Line Integrals9) Computing line integrals:
a. Evaluate
𝐼 = ∫ 𝑥𝐶 2𝑦 𝑑𝑥 + (𝑥 − 2𝑦) 𝑑𝑦
over the part of the parabola y = x2 from (0, 0) to (1, 1).
b. Same integral as previous example except C is the straight line from (0, 0) to (1, 1).
