8.1 First-Order Differential Equations

A first-order equation of the form

(8.3)

can be solved by separation of variables. We rearrange the equation to

(8.4)

and then integrate to give

(8.5)

After exponentiating the equation (making it the exponent of ), we obtain

(8.6)

The constant is determined by an initial or boundary condition.

For the inhomogeneous analog of (8.3):

(8.7)

separation of variables can be done after introducing an integrating factor. Note first that

(8.8)

Thus Eq. (8.7) can be arranged to

(8.9)

The solution is

(8.10)

This will look a lot less intimidating once the function is explicitly specified.

We will consider next a few illustrative examples of first-order differential equations of physical significance. In several cases, independent variable will be the time .

A radioactive element disintegrates at a rate which is independent of any external conditions. The number of disintegrations occurring per unit time is proportional to the number of atoms originally present. This can be formulated as a first-order differential equation in time:

(8.11)

The rate constant is a characteristic of the radioactive element. The minus sign reflects the fact that the number of atoms decreases with time. Equation (8.11) is easily solved by separation of variables. We find

(8.12)

We now evaluate the indefinite integral of each side of the equation:

(8.13)

The constant of integration is, in this case, determined by the initial condition: at time , there are atoms. Exponentiating, we obtain

(8.14)

which is the famous equation for exponential decay. Radioactive elements are usually characterized by their half-life, the time it takes for half the atoms to disintegrate. Setting , we find the relation between half-life and decay constant:

(8.15)

The result for radioactive decay is easily adapted to describe organic growth, the rate at which the population of a bacterial culture will increase given unlimited nutrition.We simply change to in Eq. (8.11), which leads to exponential growth:

(8.16)

This remains valid until the source of nutrition runs out or the population density becomes intolerable.

A bimolecular chemical reaction such as often follows a rate law

(8.17)

in which the increase in concentration of the reaction product C is proportional to the product (in the sense of ) of concentrations of the reactants A and B. To put this into more concrete form, assume that the initial concentrations at are , . At a later time , the concentration [C] grows to while [A] and [B] are reduced to . Equation (8.17) then reduces to

(8.18)

Separation of variables followed by integration gives

(8.19)

Since when , the constant equals and the solution can be written

(8.20)

More challenging is the case when , different from . If you are adventurous, you can work out the solution

(8.21)

Bernoulli’s differential equation:

(8.22)

is a well-known nonlinear equation that can be reduced to a linear equation. Multiplying by , we obtain

(8.23)

This reduces to a linear equation in the variable :

(8.24)

Using the integrating factor , the general solution, after reverting to the original variable , is given by

(8.25)

The equation is already linear if and separable, as well, if .

8.2 Numerical Solutions

Numerical methods can be used to obtain approximate solutions for differential equations that cannot be treated analytically. The most rudimentary procedure for first-order equations is the Euler method. Not, in itself, very accurate it serves as a starting point for more advanced techniques. Suppose we seek a solution of a first-order equation of the form , subject to the boundary condition . The first approximation for the value of at the point might be suggested by the starting point in the definition of a derivative

(8.26)

Choose a value for the size of each step between successive -values, , so that . Thus . Each successive step in the Euler method is then determined by the recursive relations:

(8.27)

This procedure is illustrated by the chain of red¹ segments in Figure 8.1, with the blue curve representing the hypothetical exact solution. Clearly, the accuracy degrades with increasing , although it can be improved by decreasing the interval size .

Runge-Kutta methods provide more advanced approaches to numerical integration of ordinary differential equations. The most common variant, known as RK4, is an elaboration of the Euler method using four-term interpolations to determine successive points on the solution curve. Intermediate values of the functions between and enter into the computation of successive coefficients. The RK4 method is based on the following equations:

(8.28)

with

(8.29)

(8.30)

(8.31)

(8.32)

When is a function of alone, the differential equation is essentially just an integration and the procedure reduces to Simpson’s rule. The RK4 method has an accumulated error of order , whereas the Euler method had an error which was first order in .

Mathematica is particularly adept at numerical solutions of differential equations, automatically choosing the most appropriate algorithm. For example, a first-order equation of the type discussed above can be programed using a command something like: NDSolve[{y[x]==…, y[a]==…},{x, a, b}] .

8.3 AC Circuits

A number of instructive applications of differential equations concern alternating-current (AC) circuits containing resistance, inductance, capacitance and an oscillating voltage source, as represented in Figure 8.2. In the simplest case, consider a circuit with resistance and voltage (or emf) . The current is determined by Ohm’s law:

(8.33)

The standard units are amperes for , volts for , and ohms for . The other relevant units are henrys for and farads for . Ohm’s law is true even for an AC circuit, in which the voltage varies sinusoidally with time, say

(8.34)

The current is then given by

(8.35)

Thus the current through a resistance oscillates in phase with the voltage. The frequency is expressed in units of rad/s. It is more common to measure frequency in cycles/s, a unit called the hertz (Hz). Since one cycle traces out radians, the two measures of frequency are related by

(8.36)

Thus your 60 Hz household voltage has an angular frequency of .

The voltage change across an inductance is given by . Thus for a circuit with inductance, but negligible resistance, the analog of Ohm’s law is

(8.37)

With an oscillating voltage (8.34), with an initial current , this equation is easily integrated to give

(8.38)

where the inductive reactance has the same units as . For a DC voltage , an inductor behaves just like an ordinary conductor. Note that the current in Eq. (8.38) is out of phase with the voltage . Specifically, for a pure inductance, the current lags the voltage by . Alternatively stated, the voltage leads the current by . Physically, this reflects the fact that the inductor builds up an opposing emf (by Lenz’s law) in response to an increase in current.

For a circuit with capacitance , the relevant relation is

(8.39)

where (in coulombs) is the charge on the capacitor. In a DC circuit, no current passes through a capacitor. For an AC circuit, however, with the current being given by , we find

(8.40)

For an AC voltage (8.34), the equation integrates to give

(8.41)

where the capacitive reactance is defined by . This shows that for a pure capacitance, the current leads the voltage by . The mnemonic “ELI the ICEman” summarizes the phase relationships for inductance and capacitance: for , leads while for , leads .

In an electrical circuit with both resistance and inductance, the current and voltage are related by

(8.42)

Suppose at time , while the current has the value , the voltage is suddenly turned off. With , Eq. (8.42) reduces to the form of Eq. (8.11). Thus the current must decay exponentially with

(8.43)

As a more challenging problem, let the circuit be powered by an AC voltage . Now (8.42) becomes an inhomogeneous equation.

A very useful trick when dealing with quantities having sinusoidal dependence takes advantage of Euler’s theorem (4.57) in the form

(8.44)

Note that

(8.45)

Exponentials are much simpler to differentiate and integrate than sines and cosines. Suppose the AC voltage above is imagined to have a complex form . At the end of the computation, we take the real part of the resulting equation. The answer will be the same, as if we had used throughout. (A word of caution: make certain that the complexified functions occur only linearly in the equations.)

Accordingly, we write the circuit equation

(8.46)

where is a complex variable representing the current. We can separate variables in this equation by introducing the integrating factor . Since

(8.47)

Equation (8.46) reduces to

(8.48)

Integration gives

(8.49)

If we specify that when , we find

(8.50)

The physically significant result is the real part:

(8.51)

The last term represents a transient current, which damps out for . The terms which persist represent the steady-state solution. Note that in the DC limit, as , the solution reduces to Ohm’s law . The steady-state solution can be rearranged to the form

(8.52)

Here the impedance of the RL circuit is defined by while the phase shift is given by .

The steady-state solution can be deduced directly from Eq. (8.46) by assuming from the outset that . The equation can then be readily solved to give

(8.53)

which is equivalent to (8.50). One can also define a complex impedance , in terms of which we can write a complex generalization of Ohm’s law:

(8.54)

Circuits with , , and involve second-order differential equations, which is our next topic.

8.4 Second-Order Differential Equations

We will consider here linear second-order equations with constant coefficients, in which the functions and in Eq. (8.2) are constants. The more general case gives rise to special functions, several of which we will encounter later as solutions of partial differential equations. The homogeneous equation, with , can be written

(8.55)

It is convenient to define the differential operator

(8.56)

in terms of which

(8.57)

The differential equation (8.55) is then written

(8.58)

or, in factored form,

(8.59)

where are the roots of the auxilliary equation

(8.60)

The solutions of the two first-order equations

(8.61)

give

(8.62)

while

(8.63)

gives

(8.64)

Clearly, these are also solutions to Eq. (8.59). The general solution is the linear combination

(8.65)

In the case that , one solution is apparently lost. We can recover a second solution by considering the limit:

(8.66)

(Remember that the partial derivative does the same thing as , with every other variable held constant.) Thus the general solution for this case becomes

(8.67)

When and are imaginary numbers, say and , the solution (8.65) contains complex exponentials. Since, by Euler’s theorem, these can be expressed as sums and difference of sine and cosine, we can write

(8.68)

Many applications in physics, chemistry, and engineering involve a simple differential equation, either

(8.69)

The first equation has trigonometric solutions and , while the second has exponential solutions and . These results can be easily verified by “reverse engineering.” For example, assuming that , then and . It follows that .

8.5 Some Examples from Physics

Newton’s second law of motion in one dimension has the form

(8.70)

where is the force on a particle of mass , is the acceleration, and is the velocity. Newton’s law leads to a second-order differential equation, with the solution determining the motion of the particle. The simplest case is a free particle, with . Newton’s law then reduces to

(8.71)

The solution is

(8.72)

in which and are the two constants of integration, representing, respectively, the initial position and velocity. Uniform linear motion at constant velocity is also in accord with Newton’s first law of motion: a body in motion tends to remain in motion at constant velocity, unless acted upon by some external force.

A slightly more general problem is motion under a constant force. An example is a body in free fall in the general vicinity of the Earth’s surface, which is subject to a constant downward force . Here (about 32 ft/s²), the acceleration of gravity. Denoting the altitude above the Earth’s surface by , we obtain the differential equation

(8.73)

The factors cancel. (This is actually a very profound result called the equivalence principle: the gravitational mass of a body equals its inertial mass. It is the starting point for Einstein’s General Theory of Relativity.) One integration of Eq. (8.73) gives

(8.74)

where the constant of integration represents the initial velocity. A second integration gives

(8.75)

where the second constant of integration represents the initial altitude. This solution is consistent with the well-known result that a falling body (neglecting air resistance) goes about 16 ft in 1 s, 64 ft after 2 s, and so on.

The force exerted by a metal spring subject to moderate stretching or compression is given approximately by Hooke’s law

(8.76)

Here represents the displacement, positive or negative, from the spring’s equilibrium extension, as shown in Figure 8.3. The force constant is a measure of the spring’s stiffness. The minus sign reflects the fact that the force is in the opposite direction to the displacement—a spring will resist equally either stretching and compression. Consider now a mass connected to a spring with force constant . Assume for simplicity that the mass of the spring itself is negligible compared to . Newton’s second law for this idealized system leads to the differential equation

(8.77)

The auxilliary equation (8.60) is

(8.78)

with roots

(8.79)

Thus the solution is a linear combination of complex exponentials

(8.80)

Alternatively,

(8.81)

This shows that the spring oscillates sinusoidally with a natural frequency. Sinusoidal oscillation is also called harmonic motion and the idealized system is referred to as a harmonic oscillator. If necessary, the two constants of integration can be determined by the initial displacement and velocity of the oscillating mass.

Another form for the solution (8.81) can be obtained by setting . We find then

(8.82)

An alternative possibility is .

The oscillation of a real spring will eventually be damped out, in the absence of external driving forces. A reasonable approximation for damping is a force retarding the motion which is proportional to the instantaneous velocity: , where is called the damping constant. The differential equation for a damped harmonic oscillator can be written

(8.83)

where . The auxilliary equation has the roots so that the general solution can be written:

(8.84)

For cases in which damping is not too extreme, so that , the square roots are imaginary. The solution (8.84) can be written in the form

(8.85)

This represents a damped sinusoidal wave, as shown in Figure 8.4. For stronger damping, such that , decreases exponentially with no oscillation.

A damped Hooke’s-law system subject to forced oscillations at frequency can be represented by an inhomogeneous differential equation:

(8.86)

Again it is useful to assume a complex exponential driving force . If we seek, as before, just the steady-state solutions to (8.86), the complex form can be substituted for . This reduces (8.86) to an algebraic equation for :

(8.87)

with

(8.88)

The steady-state solution is then given by

(8.89)

or, more compactly,

(8.90)

where

(8.91)

It can be seen that, after transients die out, the forced oscillation impresses its frequency on the system, apart from a phase shift . If this frequency matches the natural frequency of the oscillator, , the system is said to be in resonance. The amplitude of the oscillation then reaches a maximum. At resonance, , so that the oscillator’s motion is out of phase with the forcing function.

The behavior of an RLC circuit (Figure 8.2) is closely analogous to that of an oscillating spring. The circuit equation (8.42) generalizes to

(8.92)

Taking the time derivative leads to a more useful form:

(8.93)

recalling that . To obtain the steady-state solution, we assume the complex forms and . Equation (8.93) then reduces to an algebraic equation

(8.94)

The result is most compactly expressed in terms of the impedance:

(8.95)

Here is the reactance of the circuit, equal to the difference between inductive and capacitive reactance:

(8.96)

In terms of the complex impedance

(8.97)

Equation (8.94) can be solved for

(8.98)

Therefore the physical value of the current is given by

(8.99)

The resonance frequency for an RLC circuit is determined by the condition

(8.100)

At resonance, , , and . The inductive and capacitive reactances exactly cancel so that the impedance reduces to a pure resistance. Thus the current is maximized and oscillates in phase with the voltage. In a circuit designed to detect electromagnetic waves (e.g. radio or television signals) of a given frequency, the inductance and capacitance are “tuned” to satisfy the appropriate resonance condition.

8.6 Boundary Conditions

Often, the boundary conditions imposed on a differential equation determine significant aspects of its solutions. We consider two examples involving the one-dimensional Schrödinger equation in quantum mechanics:

(8.101)

It will turn out that boundary conditions determine the values of , the allowed energy levels for a quantum system. A particle-in-a-box is a hypothetical system with a potential energy given by

(8.102)

Because of the infinite potential nergy, the wavefunction outside the box, where or . This provides two boundary conditions

(8.103)

for the Schrödinger eqution inside the box, which can be written in the familiar form

(8.104)

where

(8.105)

The general solution to (8.104) can be written

(8.106)

Imposing the boundary condition at

(8.107)

which reduces the general solution to

(8.108)

The boundary condition at implies

(8.109)

We can’t just set because that would imply everywhere. Recall, however, that the sine function periodically goes through 0, when its argument equals Therefore the second boundary condition can satisfied if

(8.110)

This implies that and, by (8.105), the allowed values of the energy are then given by

(8.111)

where the integer is called a quantum number. Quantization of energy in a bound quantum system is thus shown to be a consequence of boundary conditions imposed on the Schrödinger equation. The wavefunction corresponding to the energy is given by

(8.112)

Setting , we obtain wavefunctions fulfilling the normalization condition:

(8.113)

8.7 Series Solutions

A very general method for obtaining solutions to second-order differential equations is to expand in a power series and then evaluate the coefficients term by term. We will illustrate the method with a trivial example which we have already solved, namely the equation with constant coefficients:

(8.121)

Assume that can be expanded in a power series about :

(8.122)

The first derivative is given by

(8.123)

We have redefined the summation index in order to retain the dependence on . Analogously,

(8.124)

Equation (8.121) then implies

(8.125)

Since this is true for all values of , every quantity in square brackets must equal zero. This leads to the recursion relation

(8.126)

Let be treated as one constant of integration. We then find

(8.127)

It is convenient to rewrite the coefficient as . We find thereby

(8.128)

The general power-series solution of the differential equation is thus given by

(8.129)

which is recognized as the expansion for

(8.130)

We consider next the more general case of a linear homogeneous second-order differential equation with nonconstant coefficients:

(8.131)

If the functions and are both finite at , then is called a regular point of the differential equation. If either or diverges as , then is called a singular point. If both and have finite limits as , then is called a regular singular point or nonessential singularity. If either of these limits continues to diverge, the point is an essential singularity.

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power-series expansion:

(8.132)

The derivatives are then given by

(8.133)

and

(8.134)

The possible values of are obtained from the indicial equation, which is based on the presumption that is the first nonzero coefficient in the series (8.132).

8.8 Bessel Functions

In later work we will encounter Bessel’s differential equation:

(8.135)

one of the classics of mathematical physics. Bessel’s equation occurs, in particular, in a number of applications involving cylindrical coordinates. Dividing the standard form (8.135) by shows that is a regular singular point of Bessel’s equation. The method of Frobenius is thus applicable. Substituting the power-series expansion (8.132) into (8.135), we obtain

(8.136)

This leads to the recursion relation

(8.137)

Setting in the recursion relation and noting that ( is the first nonvanishing coefficient), we obtain the indicial equation

(8.138)

The roots are . With the choice , the recursion relation simplifies to

(8.139)

Since , (assuming ). Likewise , as do all odd . For even , we have

(8.140)

For , the coefficients can be represented by

(8.141)

From now on, we will use the compact notation

(8.142)

Setting , we obtain the conventional definition of a Bessel function of the first kind:

(8.143)

The first three Bessel functions are plotted in Figure 8.5. Their general behavior can be characterized as damped oscillation, qualitatively similar to that in Figure 8.4.

Bessel functions can be generalized for noninteger index , as follows:

(8.144)

The general solution of Bessel’s differential equation for noninteger is given by

(8.145)

For integer , however,

(8.146)

so that is not a linearly independent solution. Following the strategy used in Eq. (8.158), we can construct a second solution by defining

(8.147)

In the limit as approaches an integer , we obtain

(8.148)

This defines a Bessel function of the second kind (sometimes called a Neumann function and written ). The computational details are horrible, but fortunately, mathematicians have worked them all out for us and these functions have been extensively tabulated. Figure 8.6 shows the first three functions .

The limiting behavior of the as is apparent from the leading term . Using the definition (8.148) we find

(8.149)

Figure 8.6 shows the logarithmic singularities as .

8.9 Second Solution

We have encountered two cases in which one solution of a second-order differential equation is relatively straightforward, but the second solution is more obscure. There is a systematic procedure for determining the second solution once the first is known.

Recall that two functions and are linearly independent if the relation

(8.150)

can only be fulfilled when . A test for linear independence is that the Wronskian, , of the two functions is not equal to zero:

(8.151)

If we did have , then , which would imply that , thus negating linear independence.

Assume that and are linearly independent solutions to the second-order differential equation (8.131). We can then write

(8.152)

Multiplying the first equation by , the second by and subtracting, we find

(8.153)

The second bracket is recognized as the Wronskian , while the first bracket equals . Thus

(8.154)

Separation of variables gives

(8.155)

Thus

(8.156)

But

(8.157)

which can be solved for to give

(8.158)

For example, suppose we know one solution for . Noting that , we can find the second solution

(8.159)

Again, for the differential equation or, more explicitly,

(8.160)

there is one obvious solution, . Noting that , the second solution follows from

(8.161)

The second solution of Bessel’s equation (8.135) for integer can be found from (8.158) with :

(8.162)

An expansion for can be obtained after calculating a power series for .

8.10 Eigenvalue Problems

An operator, designated here by , represents an action which transforms one function into another function. For the case of functions of a single variable:

(8.163)

We have already encountered the differential operator in Section 8.3. In a sense, an operator is a generalization of the concept of a function, which transforms one number into another. For the Schrödinger equation in Eq. (8.101), one can define the Hamiltonian operator by

(8.164)

The Schrödinger equation can then be written more compactly as

(8.165)

where E is the energy of the quantum system. As we have seen in Section 8.6, the boundary conditions of the problem determine a set of allowed energies and corresponding wavefunctions . These are known as eigenvalues and eigenfunctions, respectively. In pure English these are called “characteristic values” and “characteristic functions,” respectively. The corresponding German terms are eigenwert and eigenfunktion. Current usage employs a hybrid of the English and German words, namely eigenvalue and eigenfunction.

All operators in quantum mechanics which correspond to observable quantities must be Hermitian, such that

(8.166)

where and obey the same analyticity and boundary conditions that are imposed on the eigenfunctions.

The allowed set of eigenvalues and eigenfunctions obtained for some Schrödingier equation can be summarized symbolically by

(8.167)

For another solution, we can write

(8.168)

Now multiply the first equation by and the complex conjugate of the second equation by . Then subtract the two expressions and integrate over . The result can be written

(8.169)

But by the presumed Hermitian property of H, the left-hand side equals zero. Thus

(8.170)

If , the second factor becomes , which is nonzero. Therefore the first factor must equal zero, meaning that

(8.171)

Thus the energy eigenvalues must be real numbers, which is reasonable, given that they are measurable physical quantities.

Next consider the case when . Then it must be the second factor in Eq. (8.170) that equals zero:

(8.172)

Thus eigenfunctions belonging to unequal eigenvalues are orthogonal. There remains the case that for some , . The eigenfunctions and are then said to be degenerate. But it is always possible to construct linear combinations of degenerate eigenfunctions so that the orthogonality relation still applies. If the eigenfunctions are, in addition, all normalized, we obtain the compact orthonormality condition

(8.173)

The eigenfunctions then constitute an orthonormal set.