1.  Show that the first-order differential equation

(6x2y2 + sin x sin y) dx + (2y - cos x cos y + 1 + 4x3y) dy = 0,

is exact, and hence find the general solution of this differential equation.

2.  Consider the second-order non-homogeneous Cauchy-Euler differential equation

2x2  d2y + 5xdy - 9y =  10

(a)  Show that the complementary solution (of the corresponding homogeneous differential

equation) is yc (x) = C1 Vx3 + C2

(b) Use variation of parameters to show that a particular solution of the non-homogeneous

differential equation is yp (x)  =  - , and write the general solution of the non-

homogeneous differential equation.

3. We solve the following second-order differential equation

d2y           dy

dx2              dx

using a Taylor series expansion about the ordinary point x = 0.

(a)  Show that the coefficients must satisfy the recurrence formula

(2n - 3)an - n(n + 1)an+1

(b) Write out the series solution up to order x4 .

4.  Consider the differential equation with 2π-periodic non-homogeneous term

(a)  Show that the Fourier series of the non-homogeneous term f (x) is

f (x) ~  +     .

n even

[Hint:      x sin(nx)dx =  sin(nx) - x cos(nx).]

(b) What is the value of the Fourier series at x = π ?

(c) Find a particular solution yp (x) to the differential equation.

5.  Consider the following boundary value problem

x2   + x(1 - x2 )  + λex2 /2y = 0,              y(1) = 0,              y(2) = 0.

(a)  Convert the differential equation into Sturm-Liouville form.

(b) What is the inner product  (on the space of functions that satisfy the boundary

conditions) under which the eigen-solutions are mutually orthogonal ?

6.  Consider the following partial differential equation

4x2   + u  +  = 0.

(a)  By scaling x → ka x, t → kbt and u → kcu, show that η = x/Vt and φ = Vt u are

suitable similarity variables.

(b) By making use of the similarity variables η = x/Vt and φ = Vt u, show that the

partial differential equation can be converted into the following ordinary differential equation

(1 + η4 )F一一 (η) + F一 (η)F (η) + 5η3 F一 (η) + 3η2 F (η) = 0.

7.  Consider the initial-boundary value problem

=        + 2      + u,

u(0, t) = 0,    u(π, t) = 0,    -t > 0

u(x,0) = e →x ,    0 < x < π.

(a)  By separation of variables u(x, t) = X(x)T (t), obtain the following pair of ordinary

differential equations

X一一 + 2X一 + (λ + 1)X = 0,                         T˙ = -λT.

(b) Find the general solution to the differential equation for X(x) in the case of oscillatory

solutions (λ > 0).

(c) Using the boundary conditions, show that the eigen-values are λn  = n2  with n = 1, 2, 3, . . . and that the corresponding eigen-functions are Xn (x) = e →x sin(nx).

(d)  Solve the differential equation for T (t).

(e) Write down the general solution to the partial differential equation.      (f) Finally, show that the solution to the initial-boundary value problem is

o

u(x, t) =   e → (x+n2 t) sin(nx).

n odd