Families of Solutions (Level 1) The study of differential equations is in
some respect similar to that of your calculus II (integral calculus) course. As a refresher,
recall that when you were evaluating an anti-derivative or indefinite integral, we had to remember
to include the constant C of integration. An antiderivative that contains the constant
of integration represents an entire family of antiderivatives, one antiderivative for
each unique value of the constant C. For example if we were asked to find the antiderivative
or indefinite integral of the quantity 3x^2-1 dx, we would obtain the following antiderivative.
x^3 - x + C
This antiderivative represents a family of functions and is usually referred to as the general
antidervative of the function 3x^2-1. This general antiderivative represents the family
of all antiderivatives of the function 3x^2-1, notice that each antiderivative will differ
by a constant, this means that the graphs of any two antiderivatives of the function
3x^2-1 are vertical translations of each other. If we were to graph the antiderivative with
constant of integration equal to 0 we would obtain the following curve, in the same manner
if we let the constant of integration to be equal to 1, negative 1, 2 negative 2, 3 and
negative 3 we would obtain the following curves. All of these curves represent the various
antiderviatives of the function 3x^2-1, if you were to take the derivative of any of
these curves you would obtain the function 3x^2-1, as a result these curves are all
part of one happy family that can be represented by the general antiderivative x^3-x+C.
In this sense an indefinite integral contains infinitely many antiderivatives corresponding
to the unlimited number of choices that the constant of integration C can attain.
A similar concept exist when solving differential equations, for example when solving first-order
differential equations, we usually obtain a solution containing a single arbitrary constant
which we will call the parameter c. A solution containing an arbitrary constant represents
a set of solutions called a one-parameter family of solutions. For example the following
function y = ce^x represents a one-parameter family of solutions to the differential equation
y’ - y = 0 on the interval negative infinity to positive infinity, it’s a one parameter
family of solutions because it contains one constant, in this case c. In general,
when solving an nth-order differential equation, we seek an n-parameter family of solutions.
As a result a single differential equation can possess an infinite number of solutions
corresponding to the unlimited number of choices for the parameters. As an additional, example
the following function y = c sub 1 times e^x + c sub 2 times e^-x,
represents a two-parameter family of solutions to the differential equation
y’’ - y = 0, it’s a two parameter family of solutions because it contains two constants
c sub 1 and c sub 2. Now let’s talk about the various types of solutions that exist
when solving ODE’s. A solution of a differential equation that
is free of arbitrary parameters is called a particular solution. For example if we were
to take the one-parameter family of solutions y = ce^x and assign values to the parameter
c we would obtain the following solution curve. If the parameter c = 0 we would obtain the
following graph, notice that this graph represents the trivial solution y = 0 to the differential
equation y’ - y = 0, in the same manner we would obtain the following graphs when c = 1,
-1, 2, -2, 3, and -3. Each of these graphs are particular solutions of the family y = ce^x
on the interval negative infinity to positive infinity. Keep in mind that a particular solution
is free of arbitrary parameters. In the same manner the two parameter family of solutions
also contains infinitely many particular solutions such as the trivial solutions y = 0 when both
c sub 1 and c sub 2 equals 0. y = e^x when c sub 1 equals 1 and c sub 2 equals 0,
the soltuion y = e^-x
when c sub 1 equals 0 and c sub 2 equals 1, the solution y = 5e^x-2e^-x when c sub 1 equals 5 and c
sub 2 equals -2, and so on. A family of solutions containing n essential
parameters is called a general solution. Where n is the order of the differential equation
(roughly speaking, the solution of an nth order differential equation requires n integrations
and thus results in n parameters. For linear ODE’s, every possible particular
solution can be found by appropriate selection of parameters of the general solution; thus
the general solution is also a complete solution. This is not the case for nonlinear ODE’s.
For nonlinear ODE’s, (with the exception of some first-order equations) are usually
difficult or impossible to solve in terms of elementary functions. If we happened to
find a family of solutions or general solution, it is not obvious whether this family contains
all solutions. Because of the nature of nonlinear ODE’s,
a nonlinear differential equation often has a solution that cannot be obtained from the
general solution. This solution is called a singular solution. For example the solution
y = 2/sqrt(x+c) is a general solution of the differential equation 8y’ + y^3 = 0 on the interval
negative c exclusive to positive infinity, notice that the solution y = 0 is also a solution
of this differential equation, but y = 0 can’t be obtained by assigning a value to the parameter
c from the general solution. This means that the solution y = 0 is a singular solution
of the nonlinear differential equation. Note that the solution y = 0 is also a trivial
solution of the differential equation. When dealing with nonlinear ODE’s be on the lookout
for singular solutions that cannot be obtained from the general solution. They might the
trivial solution but this is not always the case.
Lastly, a solution to a differential equation can be a piece-wise defined function.
For example the family y = cx^2 is a general solution of the differential equation xy’-2y = 0 on
the interval negative infinity to positive infinity, Using the general solution we can
generate two particular solutions, the first is when we set the parameter c equal to 1,
and second is when we set the parameter c equal to negative 1. The solution curve of
each graph is represented by the following curves. It turns out that we can actually
construct a solution by using parts of each particular solution, we can essentially use
the function y = -x^2 from negative infinity to 0 exclusive in other words the values of
x are less than zero, in combination with the function y = x^2 from 0 inclusive to positive
infinity. In other words the values of x are greater than or equal to 0. This piece wise
defined function can be considered as a particular solution to the differential equation.
Note that this piece wise defined function is different from both the solution y = x^2 and y = -x^2
when looked at individually, notice that this piece wise defined function cannot be obtained
from the family y = cx^2, by a single choice of the parameter c. Rather the solution is
constructed from the family by choosing c = -1 for x < 0 and c = 1 for x >= 0. When constructing
piece wise defined solutions of differential equations, the piece wise defined function
must be differentiable and satisfy the differential equation at the given interval of definition.
Just like an explicit or implicit solution of a differential equation.
Alright to sum up, the process of solving differential equations yields solutions that
contain arbitrary parameters. When solving an nth-order differential equation, we seek
an n-parameter family of solutions. A family of solutions containing n essential parameters
is called a general solution. A solution of a differential equation that is free of arbitrary
parameters is called a particular solution. A solution that can’t be obtained by assigning
a value to the parameters c in the general solution is referred to as a singular solution.
and lastly, a solution to a differential equation can be a piece-wise defined function by restricting
the domain of two or more particular solutions. Alright in our next video will practice verifying
family of solutions of differential equations.