This tutorial document provides 4 partial differential equations and asks the reader to determine the order and linearity of each equation. It also asks the reader to find the values of m that satisfy two ordinary differential equations, where y is defined as a function of m. Specifically, it asks the reader to find the values of m such that y=xm and y=emx cos(nx) are solutions to two given ODEs.

1. TUTORIAL 1
1. Determine the order and linearity of the following DE.
d3y d4y
(a) = 1+
dx 3 dx 4
∂3 t ∂2 u ∂4 t ∂u
(b) + − + − sv = ut
∂s 3
∂v 2
∂v 4 ∂v
d4y d2u
(c) + + u = xy
dx 4 dx 2
∂2 u ∂3 w
(d) 2
− 3
+ w = y3u 2
∂v ∂y
2. Find the values of m so that y = xm is a solution of
d2y dy
x2 2
- 3x - 5y = 0
dx dx
3. Find the values of m and n if y = emx cos nx is a solution of
d2y 2dy
2
+ 2y =
dx dx