5.7 Applications to differential equations (微分方程中的应用)

本文为《Linear algebra and its applications》的读书笔记

Applications to differential equations

In many applied problems, several quantities are varying continuously in time, and they are related by a system of differential equations:

Here $x_1,...,x_n$ are differentiable functions(可导函数) of $t$ , with derivatives $x_1^{\prime },...,x_n^{\prime }$ , and the $a_{ij}$ are constants. The crucial feature of this system is that it is linear. To see this, write the system as a matrix differential equation

$${\mathbf{x}}^{\prime }(t)=A\mathbf{x}(t)(1)$$where

A solution of equation (1) is a vector-valued function that satisfies (1) for all $t$ in some interval of real numbers, such as $t\ge 0$.

Equation (1) is linear because both differentiation of functions(方程求导) and multiplication of vectors by a matrix are linear transformations. Thus, if $\mathbf{u}$ and $\mathbf{v}$ are solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$, then $c\mathbf{u}+d\mathbf{v}$ is also a solution, because

(Engineers call this property $superposition$(解的叠加) of solutions.) Also, the identically zero function is a (trivial) solution of (1).

The set of all solutions of (1) is a $subspace$ of the set of all continuous functions with values in ${\mathbb{R}}^n$.

Standard texts on differential equations show that there always exists what is called a fundamental set of solutions(基础解系) to (1). If $A$ is $n\times n$, then there are $n$ linearly independent functions in a fundamental set, and each solution of (1) is a unique linear combination of these $n$ functions. That is, a fundamental set of solutions is a basis for the set of all solutions of (1), and the solution set is an $n$-dimensional vector space of functions.

If a vector ${\mathbf{x}}_0$ is specified, then the initial value problem(初值问题) is to construct the (unique) function $\mathbf{x}$ such that ${\mathbf{x}}^{\prime }=A\mathbf{x}$ and $\mathbf{x}(0)={\mathbf{x}}_0$.

When $A$ is a diagonal matrix, the solutions of (1) can be produced by elementary calculus. For instance, consider

that is,

The system (2) is said to be $decoupled$(解耦) because each derivative of a function depends only on the function itself. From calculus, the solutions of (3) are $x_1(t)=c_1{e}^{3t}$ and $x_2(t)=c_2{e}^{?5t}$ , for any constants $c_1$ and $c_2$. Each solution of equation (2) can be written in the form

This example suggests that for the general equation ${\mathbf{x}}^{\prime }=A\mathbf{x}$, a solution might be a linear combination of functions of the form

$$\mathbf{x}(t)=\mathbf{v}{e}^{\lambda t}(4)$$for some scalar $\lambda$ and some fixed nonzero vector $\mathbf{v}$.

Observe that

Since $e^{\lambda t}$ is never zero, ${\mathbf{x}}^{\prime }(t)$ will equal $A\mathbf{x}(t)$ if and only if $\lambda \mathbf{v}=A\mathbf{v}$, that is, if and only if $\lambda$ is an eigenvalue of $A$ and $\mathbf{v}$ is a corresponding eigenvector. Thus each eigenvalue–eigenvector pair provides a solution (4) of ${\mathbf{x}}^{\prime }=A\mathbf{x}$. Such solutions are sometimes called $\mathbf{e}\mathbf{i}\mathbf{g}\mathbf{e}\mathbf{n}\mathbf{f}\mathbf{u}\mathbf{n}\mathbf{c}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{s}$(特征函数) of the differential equation. Eigenfunctions provide the key to solving systems of differential equations.

EXAMPLE 1
The circuit in Figure 1 can be described by the differential equation

where $x_1(t)$ and $x_2(t)$ are the voltages across the two capacitors at time $t$ . Suppose resistor $R_1$ is 1 ohm, $R_2$ is 2 ohms, capacitor $C_1$ is 1 farad, and $C_2$ is .5 farad, and suppose there is an initial charge of 5 volts on capacitor $C_1$ and 4 volts on capacitor $C_2$. Find formulas for $x_1(t)$ and $x_2(t)$ that describe how the voltages change over time.
SOLUTION
Let $A$ denote the matrix displayed above. For the data given,

The eigenvalues of $A$ are ${\lambda }_1=?.5$ and ${\lambda }_2=?2$, with corresponding eigenvectors

The eigenfunctions

both satisfy ${\mathbf{x}}^{\prime }=A\mathbf{x}$, and so does any linear combination of ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$. Set

and note that $\mathbf{x}(0)=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2$.

leads easily to $c_1=3$ and $c_2=?2$. Thus the desired solution of the differential equation is

Figure 2 shows the graph, or trajectory, of $\mathbf{x}(t)$, for $t\ge 0$, along with trajectories for some other initial points.

In Figure 2, the origin is called an attractor(吸引子), or sink(汇), of the dynamical system because all trajectories are drawn into the origin. The direction of greatest attraction is along the trajectory of the eigenfunction ${\mathbf{x}}_2$ (along the line through $0$ and ${\mathbf{v}}_2$ corresponding to the more negative eigenvalue, $\lambda =?2$). Trajectories that begin at points not on this line become asymptotic(渐近的) to the line through $0$ and ${\mathbf{v}}_1$ because their components in the ${\mathbf{v}}_2$ direction decay so rapidly.

If the eigenvalues in Example 1 were positive instead of negative, the corresponding trajectories would be similar in shape, but the trajectories would be traversed away from the origin. In such a case, the origin is called a repeller(排斥子), or source(源), of the dynamical system, and the direction of greatest repulsion is the line containing the trajectory of the eigenfunction corresponding to the more positive eigenvalue.

Decoupling a Dynamical System 解耦动力系统

When $A$ is diagonalizable, suppose the eigenfunctions for $A$ are

with ${\mathbf{v}}_1,...,{\mathbf{v}}_n$ linearly independent eigenvectors. Let $P=[{\mathbf{v}}_1...{\mathbf{v}}_n]$, and let $D$ be the diagonal matrix with entries ${\lambda }_1,...,{\lambda }_n$, so that $A=PD{P}^{?1}$. Now make a change of variable, defining a new function $\mathbf{y}$ by

$\mathbf{y}(t)$ is the coordinate vector of $\mathbf{x}(t)$ relative to the eigenvector basis

Since $P$ is a constant matrix, the left side of (5) is $P{\mathbf{y}}^{\prime }$. Left-multiply both sides of (5) by $P^{?1}$ and obtain ${\mathbf{y}}^{\prime }=D\mathbf{y}$, or

The change of variable from $\mathbf{x}$ to $\mathbf{y}$ has decoupled the system of differential equations, because the derivative of each scalar function $y_k$ depends only on $y_k$.

Since $y_1^{\prime }={\lambda }_1{y}_1$, we have $y_1(t)=c_1{e}^{{\lambda }_{1}t}$ , with similar formulas for $y_2,...,y_n$. Thus

To obtain the general solution $\mathbf{x}$ of the original system, compute

This is the eigenfunction expansion constructed as in Example 1.

Complex Eigenvalues

In the next example, a real matrix $A$ has a pair of complex eigenvalues $\lambda$ and $\overline{\lambda }$, with associated complex eigenvectors $\mathbf{v}$ and $\overline{\mathbf{v}}$. So two solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$ are

It can be shown that ${\mathbf{x}}_2(t)=\overline{{\mathbf{x}}_1(t)}$ by using a power series representation for the complex exponential function.

The real and imaginary parts of ${\mathbf{x}}_1$ are (real) solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$, because they are linear combinations of the solutions in (6):

To understand the nature of $Re(\mathbf{v}{e}^{\lambda t})$, recall from calculus that for any number $x$, the exponential function $e^x$ can be computed from the power series:

This series can be used to define $e^{\lambda t}$ when $\lambda$ is complex

By writing $\lambda =a+bi$ (with $a$ and $b$ real), and using similar power series for the cosine
and sine functions, one can show that

Hence

So two real solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$ are

It can be shown that ${\mathbf{y}}_1$ and ${\mathbf{y}}_2$ are linearly independent functions (when $b?=0$).

Since ${\mathbf{x}}_2(t)$ is the complex conjugate of ${\mathbf{x}}_1(t)$, the real and imaginary parts of ${\mathbf{x}}_2(t)$ are ${\mathbf{y}}_1(t)$ and $?{\mathbf{y}}_2(t)$, respectively. Thus one can use either ${\mathbf{x}}_1(t)$ or ${\mathbf{x}}_2(t)$, but not both, to produce two real linearly independent solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$.

EXAMPLE 3
The circuit in Figure 4 can be described by the equation

Suppose $R_1$ is 5 ohms, $R_2$ is .8 ohm, $C$ is .1 farad, and $L$ is .4 henry. Find formulas for $i_L$ and $v_C$ , if the initial current through the inductor is 3 amperes and the initial voltage across the capacitor is 3 volts.

SOLUTION
For the data given,

The complex solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$ are complex linear combinations of

Hence,

The real and imaginary parts of ${\mathbf{x}}_1$ provide real solutions:

Since ${\mathbf{y}}_1$ and ${\mathbf{y}}_2$ are linearly independent functions, they form a basis for the two-dimensional real vector space of solutions of ${\mathbf{x}}^{\prime }=A\mathbf{x}$. Thus the general solution is

To satisfy $\mathbf{x}(0)=\left[\begin{array}{c}3\\ 3\end{array}\right]$,

or

See Figure 5.

In Figure 5, the origin is called a spiral point(螺旋极点) of the dynamical system. The rotation is caused by the sine and cosine functions that arise from a complex eigenvalue. The trajectories spiral inward because the factor $e^{?2t}$ tends to zero. Recall that $?2$ is the real part of the eigenvalue in Example 3. When $A$ has a complex eigenvalue with positive real part, the trajectories spiral outward. If the real part of the eigenvalue is zero, the trajectories form ellipses around the origin.