HIT223Worksheet IV
Engineering Mathematics IV
1. Find the general solution to each of the following partial differential equation, where u = u(x, y).
(a)
∂u 1
− u = 0,
∂y
y
(b)
∂ 2 u 1 ∂u
−
= 0,
∂y 2
y ∂y
(c)
∂2u
1 ∂u
+
= x,
∂x∂y y ∂x
y 6= 0
y 6= 0
y 6= 0
2. Show that the change of variables v = x2 − y and w = x2 + y reduces the equation
2
∂ 2 u 1 ∂u
2∂ u
−
= 0,
−
4x
∂x2
x ∂x
∂y 2
x 6= 0
to
∂2u
=0
∂v∂w
Hence find the general solution of the equation.
3. Use the change of variables
v = x2 + y 2 ,
w=y
to find the general solution of the differential equation
y2
2
∂2u
y 2 ∂u x2 ∂u
∂2u
2∂ u
−
2xy
+
x
−
−
= 0,
∂x2
∂x∂y
∂y 2
x ∂x
y ∂y
x 6= 0,
4. Solve the following boundary value problems.
(a)
∂2u
∂u
=
2
∂x
∂y
given the boundary conditions
u(x, 0) = 6e−4y and
(b)
∂u
∂u
=
given that u(x, 0) = 4e−3x
∂x
∂x
1
∂u
(0, y) = 4e−4y
∂x
y 6= 0