This section contains information about
A linear second-order ODE has the form:
On any interval where S(t) is not equal to 0, the above equation
can be divided by S(t) to yield
The equation is called homogeneous if f(t)=0. Otherwise, it is
called
nonhomogeneous.
A second-order differential equation is accompanied by initial
conditions
or boundary conditions. Initial conditions are in
the form y(t_0)=y_0 and
y'(t_0)=y'_0. Boundary conditions
might be of the form: y(t_0)=a and y(t_1)=b.
For the initial value problem, the existence and uniqueness
theorem states
that if p(t), q(t) and f(t) are continuous on
some interval (a,b) containing t_0,
then there exists a unique solution
y(t) to the ode in the whole interval (a,b).
Procedure for Solving Linear Second-Order ODE
The procedure for solving linear second-order ode has two steps
According to the theory for linear differential equations, the
general
solution of the homogeneous problem is
where C_1 and C_2 are constants and y_1 and y_2 are any two
linearly independent solutions to the homogeneous equation.
The particular solution is any solution of the nonhomogeneous
problem and is denoted y_p(t).
The general solution of the full nonhomogeneous problem is
The key point to note is that all possible solutions
to a linear second-order
ode can be obtained from two
linearly independent solutions to the
homogeneous problem
and any particular solution.
Here is an example. Consider the ode
The homogeneous equation is
It can be shown that y_1=exp(-t) and y_2=exp(-2t) are solutions
to the
homogeneous equation. Plug these expressions into the
ode and verify!
A particular solution of the nonhomogeneous equation is exp(t).
Hence,
the general solution of the ode is
where C_1 and C_2 are constants.
Two functions are linearly independent if they are not multiples
of each other.
For example, exp(-t) and exp(-2t) are linear
independent. On the other
hand, t+3 and 7t+21 are linearly
dependent since the latter is 7 times the former.
Techniques for Solving Homogeneous Linear Second-Order ODE
Certain classes of homogeneous linear second-order ode can
be solved
analytically. We will consider two classes:
p(t) and q(t) are constants.
p and q are constants.
Techniques for Determining a Particular Solution
There are two principal techniques for determining a particular solution:
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