8_differential_equations_2.md

title: Differential Equations 2

Differential equations 2

The lecture on differential equations consists of three parts, each with their own video:

• Higher order linear differential equations
• Partial differential equations: Separation of variables
• Self-adjoint differential operators

Total video length: hour minutes seconds

Higher order linear differential equations

In the previous lecture, we focused on first order linear differential equations as well as systems of such equations. In this lecture we switch focus to DE's which involve higher derivatives of the function we would like to solve for. To f`%acilitate this change we are going to change notation. In the previous lecture we wrote differential equations for x(t). In this lecture we will write DE's of y(x), where y is an unknown function and x is the independent variable. For this purpose we make the following definitions,

y' = \frac{dy}{dx}, \ y'' = \frac{d^2 y}{dx^2}, \ \cdots, \ y^{(n)} = \frac{d^n y}{dx^n}.

In the new notation, a linear n-th order differential equation with constant coefficients reads

y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0.

!!! info "Linear combination of solutions are still solutions"

Note that as was the case for first order linear DE's, the propery of 
linearity once again means that if $y_{1}(x)$ and $y_{2}(x)$ are both 
solutions, and $a$ and $b$ are constants, 

$$a y_{1}(x) + b y_{2}(x)$$

then linear combination of the solutions is also a solution.

In order to solve a higher order linear DE we will present a trick that makes it possible to map the problem of solving a single n-th order linear DE into a related problem of solving a system of n first order linear DE's.

To begin, define:

y_{1} = y, \ y_{2} = y', \ \cdots, \ y_{n} = y^{(n-1)}.

Then, the differential equation can be re-written as

y_1 ' = y_2 y_2 ' = y_3 \vdots y_{n-1} ' = y_{n} y_{n} ' = - a_{0} y_{1} - a_{1} y_{2} - \cdots - a_{n-1} y_{n}.

Notice that together these n equations form a linear first order system, the first n-1 equations of which are trivial. Note that this trick can be used to reduce any system of n-th order linear DE's to a larger system of first order linear DE's.

Since we have discussed already the method of solution for first order linear systems, we will outline the general solution to this system. As before, the general solution will be the linear combination of n linearly independent solutions f_{i}(x), i \epsilon \{1, \cdots, n \}, which make up a basis for the solution space. That is the general solution has the form

y(x) = c_1 f_1 (x) + c_2 f_2 (x) + \cdots + c_n f_{n}(x).