Basic object of study: second order linear differential equations
Initial value problem:
Basic fact: if p(t), q(t), and g(t) are continuous on an interval around t_{0}, then any initial value problem has a unique solution on that interval. Our Basic goal: find the solution!
Homogeneous: g(t) Constant coefficients: p(t) and q(t) are constant.
Start small: homogeneous with constant coefficients:
Basic idea: guess that y = e^{rt}, and plug in! Get:
Solve: get (typically) two roots r_{1}, r_{2}, so y_{1}e^{r1 t} and y_{2}e^{r2 t} are both solutions.
The equation ar^{2}+br+c = 0 is called the characteristic equation for our differential equation.
Operator notation: write L[y] = y^{¢¢}+p(t)y^{¢}+ q(t)y (this is called a linear operator), then a solution to (*) is a function y with L[y] = g(t).
For a linear differential equation, L[c_{1}y_{1}+c_{2}y_{2}] = c_{1}L[y_{1}]+c_{2}L[yt], and so if y_{1} and y_{2} are both solutions to L[y] = 0 then so is c_{1}y_{1}+c_{2}y_{2} . c_{1}y_{1}+c_{2}y_{2} is called a linear combination of y_{1} and y_{2}. This is called the Principle of Superposition: more generally, if L[y_{1}] = g_{1}(t) and L[y_{2}] = g_{2}(t), then L[y_{1}+y_{2}] = g_{1}(t)+g_{2}(t) .
With (the right) two solutions y_{1}, y_{2} to a homogeneous equation
we can solve any initial value problem, by choosing the right linear combination: we need to solve
for the constants c_{1} amd c_{2}; then y = c_{1}y_{1}+c_{2}y_{2} is our solution. This we can do directly, as a pair of linear equations, by solving one equation for one of the constants, and plugging into the other equation, or we can use the formulas

 



 



 



 


where 

 



 


W is called the Wronskian (determinant) of y_{1} and y_{2} at t_{0} . The Wronskian is closely related to the concept of linear independence of a collection y_{1},¼,y_{n} of functions; such a collection is linearly independent if the only linear combination c_{1}y_{1}+ ¼+ c_{n}y_{n} which is equal to the 0 function is the one with c_{1} = ¼ = c_{n} = 0 .
Two functions y_{1} and y_{2} are linearly independent if their Wronksian is nonzero at some point; for a pair of solutions to (**), it turns out that the Wronskian is always equal to a constant multiple of
and so is either always 0 or never 0. We call a pair of linearly independent solutions to (**) a pair of fundamental solutions. By our above discussion, we can solve any initial value problem for (**) as a linear combination of fundamental solutions y_{1} and y_{2}. By our existence and uniqueness result, this give us:
If y_{1} and y_{2} are a fundamental set of solutions to the differential equation (**), then
any solution to (**) can be expressed as a linear combination
c_{1}y_{1}+c_{2}y_{2} or y_{1} and y_{2}.
So to solve an initial value problem for (**), all we need is a pair of fundamental solutions.
For an equation with constant coefficients, we do this by finding the roots of the
characteristic equation ar^{2}+br+c = 0 . We have the following basic facts:
If the roots of the characteristic equation are real and distinct, r_{1} ¹ r_{2}, then a fundamental set of solutions is
If the root of the characteristic equation are complex a±bi, then a fundamental set of solutions is
If the roots of the characteristic equation are repeated (and therefore real), r_{1} = r_{2} = r, then a fundamental set of solutions is
In showing the last of these facts, we introduced a general technique for
finding a second, linearly independent, solution y_{2} to (**), given a (nonzero)
solution y_{1}; this was called reduction of order; if y_{1} is a solution
to (**), then so is
This formula arises by assuming that y_{2}(t) = c(t)y_{1}(t), and then determining what differential equation c(t) must satisfy! It turns out to be a firstorder equation (hence the name reduction of order).
Much of what we just did for second order equation goes through without any change for
even higher order (linear) equations:
and its associated homogeneous equation\
In this case the correct notion of an initial value problem requires us to specify the values, at t_{0}, of y and all its derivatives up to the (n1)st:
As with the second order case, we have a principle of superposition: L[y_{1}] = g_{1} and L[y_{2}] = g_{2}, then L[y_{1}+y_{2}] = g_{1}+g_{2} . This means that linear combinations of solutions to the homogeneous equation (!!) are also solutions. And the general solution to (!!) can always be obtained (uniquely) as a linear combination of n linearly independent (or fundamental) solutions. Linear independence can be determined by computing a Wronskian determinant W(y_{1},¼,y_{n}).
The theory we developed for homogeneous equations with constant coefficients can be similarly extended. The equation
has a fundamental set of solutions determined by its characteristic equation
Real roots r correspond to solutions exp(rt) ; complex roots to solutions exp(at)cos(bt) and exp(at)sin(bt) . The only extra wrinkle is that we can have repeated roots which repeat many times, and even repeated complex roots! For each, we do as we did before and create new fundamental solutions by multiplying our basic solution by t, as many times as it repeats. For example, the equation
has a characteristic equation with roots i,i,i, and i, and so its fundamental solutions are
Our final concern is inhomogeneous linear equations
with g(t) ¹ 0 . The principle of superposition tells us that for any pair of solutions Y_{1}, Y_{2} to (!), L[Y_{2}Y_{1}] = 0, and so if we have a fundamental set of solutions to the associated homogeneous equation, y_{1},¼,y_{n}, we can write
In other words, we can find any solution to (!) by finding one particular solution, together with a fundamental set of solutions to the associated homogeneous equation (!!). Any initial value problem can then be solved by solving the system of equations
all the way to
for the constants c_{1},¼,c_{n} .
The only part of this we haven't really explored yet is finding a particular solution to (!). for this we have two techniques. The first is called the Method of Undetermined Coefficients.
The basic idea behind the technique is that for most kinds of functions, like polynomials, expoential, sines and cosines, or products of these, all of the functions derivatives are of the same kind. So if the function g(t) in
is one of these kinds, what we do is guess that our solution y is the same kind. In particular,
If g is a polynomial of degree n, we set y to be a (different) polynomial
of degree n,
If g is a multiple of an exponential exp(rt), we set y to be a a multiple cexp(rt) of g,
If g is a multiple of sin(bt) or cos(bt), we set y to be a linear combination asin(bt)+bcos(bt),
If g(t) = exp(rt)cos(bt) (or has a sine), we set y to be aexp(rt)sin(bt)+bexp(rt)cos(bt),
If g is a polynomial of degree n times one of these, we set y to be a (different) polynomial of degree n time the corresponding function above.
Then we must plug this function into (!), and solve for the undetermined
coefficients.
Of course, there is one wrinkle; sometimes our choice of y cannot work, because it is a solution to the associated homogeneous equation. For example, for the equation
The function y = acos(t)+bsin(t) will never solve it, because for such a function, L[y] In this case what we must do is multiply our guess by t, or more generally, by a high enough power to insure that our guess is not a solution to the homogeneous equation. For this, we must first determine the number of times the root which corresponds to our target solution occurs among the roots of the associated characteristic equation. This can be a trifle tricky to determine; for example, for the equation
we should guess that our solution is y = t(at+b)e^{t}, since our original guess would be y = (at+b)e^{t}, but this is a solution to the homogeneous equation, while t times it is not; but for
we should guess that our solution is y = at^{2}e^{t}, since te^{t} is still a solution to the homogeneous equation, but y = t^{2}e^{t} is not.
Finally, if our function g(t) is a linear combination of such functions, we can use this method to solve L[y]each piece, and then use the Principle of Superposition to find our solution by taking a linear combination.
Our other technique works (in theory) for any linear equation; we will restrict our attention
to second order equations, for sake of simplicity. It is called variation of parameters,
and starts with a pair of fundamental solutions y_{1},yt to the associated homogeneous equation\
and then guess that the solution to our inhomogeneous equation
is of the form y(t) = c_{1}(t)y_{1}(t)+c_{2}(t)y_{2}(t), and plug in. The resulting equation is too complicated, but if we make the simplifying assumption
then the equation becomes
which we can solve:

 



 


Here again, the by now familiar Wronskian appears! Note that we must still integrate these functions, to determine c_{1} and c_{2} .