Solving the Linear equation
Mixture Problems
RL Circuits
Linear First-Order differential Equations
If a differential equation can be written in the form
,then it is a linear first-order equation in standard form.
See Example 1 - 2, pages 499 - 500.
Solving the Linear Equation
The solution of the linear first-order differential equation written in
standard form is
,where
.In the formula for
,
any antiderivative of 
will do.See Examples 3 - 4, pages 501 - 502.
Example 4 is an initial value problem.
Mixture Problems
A chemical in a solution runs into a container holding the solution with a
given amount of the chemical dissolved in it as well. The mixture is kept
uniform by stirring and flows out of the container at a known rate. It is often
important to know the concentration of the chemical in the container at any
given time. The differential equation describing the process can be written as
Rate of change rate at which rate at which
of amount = chemical - chemical
in container arrives departs
If
is the amount of chemical in the container at time 
and 
is the total volume of liquid in the container at time
,
then the departurerate of the chemical at time
is
Or
.Example 5, pages 503 - 505, shows the solution of the
linear first-order equation
,where
and 
.RL Circuits
The diagram shows an electrical circuit whose total resistance is a
constant
ohms and whose self-inductance, shown as a coil, isa constant
henries. A switch at terminals 
and 
can beclosed to connect an electrical source of constant
volts. Ohm's law,
,needs modification for this circuit.
The modified form is
,where
is the intensity of the current in amperes and 
is the timein seconds. Solving this equation, we can predict how the current will
flow after the switch is closed.
See Example 6, page 505.
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