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By Giuseppe Tenti and annotated by Douglas Wilhelm Harder.
In general, we will be discussing a single unknown function in a
single variable. For example, we may consider the function
and the variable
; the function
and the variable
; or the function
and the variable
. It unnecessarily obfuscates the mathematics
to include the variable in which the function is defined, and therefore
we will use, for example, and the variable
is implied from the context.
For other functions, the variable will always be defined. For example,
a constant voltage or resistance will be represented by a symbol,
or , but if the voltages
and resistances are variable,
or , respectively.
We we define a function, we will also use indicate the variable,
and therefore, the notation
defines the function
Solving Differential Equations (MEM 10.4)
From the previous lecture, we saw that we may use differential equations
to describe natural phenomena. For example, the circuit was described
by the differential equation
Let us consider, for example, an easier case where the differential equation
is defined by a circuit with a voltage source
, a resistor with resistance
, and a capacitor with capacitance
, all in
a single loop. This yields the differential equation:
If we assume that
and the voltage source is zero, we get the
simplified differential equation
In this case, we notice that if we define
, this is a function where
the function plus the derivative is 0 and therefore, this function satisfies
the differential equation. However, the functions
also satisfy the differential
equation. In each case, the function gives us a curve which describes
how the charge on the capacitor decreases as time grows. Figure 1
shows an entire collection of functions, all of which satisfy the
differential equation: the value of the function is always equal to
the negative of the slope.
Figure 1. Numerous solutions to the differential equation
Suppose we change the voltage source to be a constant source of
one volt. Now the differential equation is
and various solutions include
, and Figure 2 shows
an entire collection of such solutions:
Figure 2. Numerous solutions to the differential equation
If we now change the voltage source to be an oscillating function
, we get a different
class of solutions, as is shown in Figure 3.
Figure 3. Numerous solutions to the differential equation
Now, it appears that the charge on the capacitor begins
to oscillate, too, but the oscillation appears to look more
One important fact the reader may noted is that the solutions to these
three 1st-order ordinary differential equations never cross. They
may get arbitrarily close, but they do not appear to cross each other.
This is, in fact, expected but is usually only proved in an upper-level
course on differential equations.
We will conclude with two more examples. We will introduce a variable
resistor and a switch.
Suppose we now add a variable resistor
where the resistance is given by the function
as time increases.
The differential equation is now
Figure 4 shows numerous solutions to this differential equation.
Figure 4. Numerous solutions to the differential equation
The solutions decay to 0 like they did in Figure 1, but they do not
decay as quickly: the greater resistance prevents the system from
Issue: The denominator of the
indicates that the behaviour would be identical whether we have
a variable resistor with
, or a variable capacitor with
As an alternate example, suppose we have a voltage source which
is initially 0 but after half a second, that is, at time
, the voltage is switch to two volts.
Engineers use the unit step function,
to represent such
switches and it is defined as
In this case, we may represent the turning on of the switch by
the function . Thus, the differential
equation is now
and Figure 5 shows the various solutions to the differential equations.
Figure 5. Numerous solutions to the differential equation
You will note that in each case, there are many different solutions
to a differential equation, but if we start a circuit with the same
conditions at time
, we expect the response of the circuit
to be the same.
In this case, the state of the circuit defines which solution we
choose, in particular, the we look at the initial charge on the
The previous lecture has shown that given any 1st-order ODE, there exist multiple solutions. When a differential
equation is used to describe, for example, a discharging capacitor, the charge on the capacitor follows only one of these
solutions. To select one of the many solutions, we must introduce a constraint which is usually of the form
that is, we know the state of the solution at one particular moment in time and from this, we should be able to deduce
the solution for all time. To simplify the mathematics, we will usually define the point in time when the solution is
known to be , that is, we specify
Because we are specifying the state of the system at one point in time and usually we are attempting to determine
the solution from that point on, that is, for , we call the constraint an
initial value and the ordinary differential equation together with the initial value is called
an initial-value problem.
Homogeneous 1st-order Linear Ordinary Differential Equations
A homogeneous 1st-order LODE is of the form
where the function is a function of the independent variable. Suppose
we need to find a solution to this differential equation subject to the condition