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 t₀, 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 r₁, r₂, so y₁e^{r₁ t} and y₂e^{r₂ t} are both solutions.

The equation ar²+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[c₁y₁+c₂y₂] = c₁L[y₁]+c₂L[yt], and so if y₁ and y₂ are both solutions to L[y] = 0 then so is c₁y₁+c₂y₂ . c₁y₁+c₂y₂ is called a linear combination of y₁ and y₂. This is called the Principle of Superposition: more generally, if L[y₁] = g₁(t) and L[y₂] = g₂(t), then L[y₁+y₂] = g₁(t)+g₂(t) .

With (the right) two solutions y₁, y₂ to a homogeneous equation

we can solve any initial value problem, by choosing the right linear combination: we need to solve

for the constants c₁ amd c₂; then y = c₁y₁+c₂y₂ 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 | 

a	b
c	d

| = ad-bc . This makes it clear that a solution exists (i.e., we have the `right' pair of functions), provided that the quantity

W is called the Wronskian (determinant) of y₁ and y₂ at t₀ . The Wronskian is closely related to the concept of linear independence of a collection y₁,¼,y_n of functions; such a collection is linearly independent if the only linear combination c₁y₁+ ¼+ c_ny_n which is equal to the 0 function is the one with c₁ = ¼ = c_n = 0 .

Two functions y₁ and y₂ are linearly independent if their Wronksian is non-zero 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 y₁ and y₂. By our existence and uniqueness result, this give us:

If y₁ and y₂ are a fundamental set of solutions to the differential equation (**), then any solution to (**) can be expressed as a linear combination c₁y₁+c₂y₂ or y₁ and y₂.

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 ar²+br+c = 0 . We have the following basic facts:

If the roots of the characteristic equation are real and distinct, r₁ ¹ r₂, 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), r₁ = r₂ = 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 y₂ to (**), given a (non-zero) solution y₁; this was called reduction of order; if y₁ is a solution to (**), then so is

This formula arises by assuming that y₂(t) = c(t)y₁(t), and then determining what differential equation c(t) must satisfy! It turns out to be a first-order 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 t₀, of y and all its derivatives up to the (n-1)st:

As with the second order case, we have a principle of superposition: L[y₁] = g₁ and L[y₂] = g₂, then L[y₁+y₂] = g₁+g₂ . 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(y₁,¼,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 Y₁, Y₂ to (!), L[Y₂-Y₁] = 0, and so if we have a fundamental set of solutions to the associated homogeneous equation, y₁,¼,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 c₁,¼,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 = at²e^t, since te^t is still a solution to the homogeneous equation, but y = t²e^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 y₁,yt to the associated homogeneous equation\

and then guess that the solution to our inhomogeneous equation

is of the form y(t) = c₁(t)y₁(t)+c₂(t)y₂(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 c₁ and c₂ .