Brushingup Math 251 for ChE 210
1. Separable equations
Many firstorder differential equations can be reduced to the form
g(y) y’
=
f(x)
(1)
By algebraic manipulations. Since
y’ = dy / dx
, we find it convenient to write
g(y) dy
=
f(x) dx
(2)
This is merely another way of writing eq (1). Such an equation is called an
equation with
separable variables
, or a
separable equation
, because in eq (2) the variables x and y are
separated so that x appears only on the right and y appears only on the left. By integrating on
both sides of eq (2), we obtain
∫
g(y)dy =
f(x)dx + c
(3)
If
we assume that f and g are continuous functions, the integrals in eq (3) will exist, and by
evaluating these integrals we obtain the general solutions of eq (1).
2. Equations reducible to separable form
3. Exact differential equations
4. Integrating factors
5. Linear firstorder differential equations
A firstorder differential equation is said to be
linear
if it can be written
y’ + f(x) y = r(x)
(4)
The characteristic feature of this equation is that it is linear in
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 Spring '08
 SEONGKIM
 Derivative, Mass Balance, Method of variation of parameters, firstorder differential equations

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