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1. Setting the Scene
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where the constant c typically is determined by an extra condition. For
instance, if we require
u(0)=1/2,
1 t
we get c =1/2 and u(t)= e . So keep this in mind; the solution we seek
2
from a differential equation is a function.
Concepts
1.1.1
We usually subdivide differential equations into partial differential equa-
tions (PDEs) and ordinary differential equations (ODEs). PDEs involve
partial derivatives, whereas ODEs only involve derivatives with respect to
one variable. Typical ordinary differential equations are given by
(a) u (t)= u(t),

(b) u (t)= u (t),
2
(c) u (t)= u(t) + sin(t) cos(t), (1.1)

(d) u (x)+ u (x)= x ,
2

(e) u (x) = sin(x).
Here (a), (b) and (c) are “first order” equations, (d) is second order, and
(e) is fourth order. So the order of an equation refers to the highest order
derivative involved in the equation. Typical partial differential equations
are given by 1
(f) u t (x, t)= u xx (x, t),
(g) u tt (x, t)= u xx (x, t),
(h) u xx (x, y)+ u yy (x, y)=0,

(i) u t (x, t)= k(u(x, t))u x (x, t) ,
x
(1.2)
(j) u tt (x, t)= u xx (x, t) − u (x, t),
3
1
(k) u t (x, t)+ u (x, t) = u xx (x, t),
2
2 x
(l) u t (x, t)+(x + t )u x (x, t)=0,
2
2
(m) u tt (x, t)+ u xxxx (x, t)=0.
Again, equations are labeled with orders; (l) is first order, (f), (g), (h), (i),
(j) and (k) are second order, and (m) is fourth order.
Equations may have “variable coefficients,” i.e. functions not depending
on the unknown u but on the independent variables; t, x,or y above. An
equation with variable coefficients is given in (l) above.
2
1 Here u t = ∂u , u xx = ∂ u
∂t ∂x 2 , and so on.