2.1 Differential Equations
This section considers only ordinary differential equations, which are those with only one independent variable. [A partial differential equation has more than one independent variable. An example is the time-dependent Schrödinger equation (1.16), in which $t$ and $x$ are the independent variables.] An ordinary differential equation is a relation involving an independent variable $x$, a dependent variable $y(x)$, and the first, second, $\ldots, n$th derivatives of $y\left(y^{\prime}, y^{\prime \prime}, \ldots, y^{(n)}\right)$. An example is
\( \begin{equation} y^{\prime \prime \prime}+2 x\left(y^{\prime}\right)^{2}+y^{2} \sin x=3 e^{x} \tag{2.1} \end{equation} \)
The order of a differential equation is the order of the highest derivative in the equation. Thus, (2.1) is of third order.
A special kind of differential equation is the linear differential equation, which has the form
\( \begin{equation} A{n}(x) y^{(n)}+A{n-1}(x) y^{(n-1)}+\cdots+A{1}(x) y^{\prime}+A{0}(x) y=g(x) \tag{2.2} \end{equation} \)
where the $A$ 's and $g$ (some of which may be zero) are functions of $x$ only. In the $n$ th-order linear differential equation (2.2), $y$ and its derivatives appear to the first power. A differential equation that cannot be put in the form (2.2) is nonlinear. If $g(x)=0$ in (2.2), the linear differential equation is homogeneous; otherwise it is inhomogeneous. The onedimensional Schrödinger equation (1.19) is a linear homogeneous second-order differential equation.
By dividing by the coefficient of $y^{\prime \prime}$, we can put every linear homogeneous secondorder differential equation into the form
\( \begin{equation} y^{\prime \prime}+P(x) y^{\prime}+Q(x) y=0 \tag{2.3} \end{equation} \)
Suppose $y{1}$ and $y{2}$ are two independent functions, each of which satisfies (2.3). By independent, we mean that $y{2}$ is not simply a multiple of $y{1}$. Then the general solution of the linear homogeneous differential equation (2.3) is
\( \begin{equation} y=c{1} y{1}+c{2} y{2} \tag{2.4} \end{equation} \)
where $c{1}$ and $c{2}$ are arbitrary constants. This is readily verified by substituting (2.4) into the left side of (2.3):
\( \begin{align} & c{1} y{1}^{\prime \prime}+c{2} y{2}^{\prime \prime}+P(x) c{1} y{1}^{\prime}+P(x) c{2} y{2}^{\prime}+Q(x) c{1} y{1}+Q(x) c{2} y{2} \ & \quad=c{1}\left[y{1}^{\prime \prime}+P(x) y{1}^{\prime}+Q(x) y{1}\right]+c{2}\left[y{2}^{\prime \prime}+P(x) y{2}^{\prime}+Q(x) y{2}\right] \ & \quad=c{1} \cdot 0+c{2} \cdot 0=0 \tag{2.5} \end{align} \)
where the fact that $y{1}$ and $y{2}$ satisfy (2.3) has been used. The general solution of a differential equation of $n$th order usually has $n$ arbitrary constants. To fix these constants, we may have boundary conditions, which are conditions that specify the value of $y$ or various of its derivatives at a point or points. For example, if $y$ is the displacement of a vibrating string held fixed at two points, we know $y$ must be zero at these points.
An important special case is a linear homogeneous second-order differential equation with constant coefficients:
\( \begin{equation} y^{\prime \prime}+p y^{\prime}+q y=0 \tag{2.6} \end{equation} \)
where $p$ and $q$ are constants. To solve (2.6), let us tentatively assume a solution of the form $y=e^{s x}$. We are looking for a function whose derivatives when multiplied by constants will cancel the original function. The exponential function repeats itself when differentiated and is thus the correct choice. Substitution in (2.6) gives
\( \begin{gather} s^{2} e^{s x}+p s e^{s x}+q e^{s x}=0 \ s^{2}+p s+q=0 \tag{2.7} \end{gather} \)
Equation (2.7) is called the auxiliary equation. It is a quadratic equation with two roots $s{1}$ and $s{2}$ that, provided $s{1}$ and $s{2}$ are not equal, give two independent solutions to (2.6). Thus, the general solution of (2.6) is
\( \begin{equation} y=c{1} e^{s{1} x}+c{2} e^{s{2} x} \tag{2.8} \end{equation} \)
For example, for $y^{\prime \prime}+6 y^{\prime}-7 y=0$, the auxiliary equation is $s^{2}+6 s-7=0$. The quadratic formula gives $s{1}=1, s{2}=-7$, so the general solution is $c{1} e^{x}+c{2} e^{-7 x}$.