RESONANCE IN LINEAR DIFFERENTIAL EQUATIONS AND L’HOSPITAL’S RULE*

Joseph Wiener The University of Texas – Pan American

Historically, differential equations grew into an independent subject from a branch of calculus. It is very useful to cultivate this connection in both calculus and differential equations courses, all the more that today’s changes in calculus teaching will influence our approach to differential equations. From this standpoint, we would like to continue the discussion initiated in [1] on the application of L’Hospital’s rule to the study of the solutions of linear differential equations with constant coefficients.

Consider the undamped mass–spring system with a given periodic external force. The equation of motion is

(1)  x′′ + ω_{0}^{2}x = A cos ωt,

where ω_{0} is the natural frequency of the system and ω is the applied

* Research partially supported by U.S. Army Grant DAAL03– 89–G–0107. frequency. If ω ≠ ω_{0}, a particular solution

(2)  x_{p}(t) = \frac{A}{ω_{0}^{2} − ω^{2}} cos ωt

can easily be found by the method of undetermined coefficients. If ω is close to ω_{0} then x_{p} represents an oscillation with large amplitude. If we think of ω_{0} as fixed and let ω approach ω_{0}, the amplitude of the oscillation becomes unbounded. This phenomenon is called resonance. If ω = ω_{0}, a particular solution cannot be obtained from (2). However, since the numerator in (2) tends to A cos ω_{0}t as ω → ω_{0} and

x_{0}(t) = \frac{A}{ω_{0}^{2} − ω^{2}} cos ω_{0}t

is a solution of the equation x′′ + ω_{0}^{2}x = 0, it is preferable to take a particular solution of (1) in the form

(3)  x(t, ω) = x_{p}(t) − x_{0}(t) = \frac{A(cos ωt − cos ω_{0}t)}{ω_{0}^{2} − ω^{2}}.

Employing L’Hospital’s rule we find the limit of (3) as ω → ω_{0} which yields a particular solution of (1) in the case ω = ω_{0}:

lim_{ω→ω_{0}} x(t, ω) = \frac{At}{2ω_{0}} sin ω_{0}t.

Incidentally, a similar situation takes place with the formula for the power integral

∫ x^{a}dx = \frac{x^{a+1}}{a + 1} + C,

where a ≠ −1 is any real number and x > 0. Since x^{a+1} → 1 as a → −1, we write

C = −\frac{1}{a + 1} + C_{1}

and

∫ x^{a}dx = \frac{x^{a+1} − 1}{a + 1} + C_{1}.

This formula “works” also at a = −1, for L’Hospital’s rule shows that

lim_{a→−1} \frac{x^{a+1} − 1}{a + 1} = ln x.

Now consider, for example, the equation

(4)  x′′ − 2ω_{0}x′ + ω_{0}^{2}x = Ae^{ωt},

which admits a particular solution

(5)  x_{p}(t) = \frac{A}{(ω − ω_{0})^{2}} e^{ωt}, ω ≠ ω_{0}.

Again, a particular solution cannot be obtained from here at ω = ω_{0}, and in this case it is impossible to alleviate the difficulty by subtracting from (5) the solution

\frac{Ae^{ω_{0}t}}{(ω − ω_{0})^{2}}

of the homogeneous equation corresponding to (4). Indeed, ω_{0} is a double root of the denominator (ω − ω_{0})^{2} and only a simple root of the numerator A[e^{ωt} − e^{ω_{0}t}]. To accomplish the task, we expand e^{ωt} as a function of ω in a Taylor series

e^{ωt} = e^{ω_{0}t} + te^{ω_{0}t}(ω − ω_{0}) + ⋯

centered at ω_{0} and observe that

x_{0}(t) = \frac{A}{(ω − ω_{0})^{2}} [e^{ω_{0}t} + (ω − ω_{0})te^{ω_{0}t}]

is a solution of the equation x′′ − 2ω_{0}x′ + ω_{0}^{2}x = 0. Moreover, the difference

x(t, ω) = x_{p}(t) − x_{0}(t) = \frac{A}{(ω − ω_{0})^{2}} [e^{ωt} − e^{ω_{0}t} − (ω − ω_{0})te^{ω_{0}t}]

is a particular solution of equation (4) whose numerator and its derivative with respect to ω approach zero as ω → ω_{0}. By L’Hospital’s rule,

lim_{ω→ω_{0}} x(t, ω) = lim_{ω→ω_{0}} \frac{A[te^{ωt} − te^{ω_{0}t}]}{2(ω − ω_{0})} = lim_{ω→ω_{0}} \frac{At^{2}e^{ωt}}{2} = \frac{At^{2}}{2} e^{ω_{0}t},

which is a particular solution of (4) when ω = ω_{0}. This derivation reveals the nature of the solutions involved and a quick method for their construction. Assuming ω ≠ ω_{0}, we find by the method of undetermined coefficients solution (5). The sum S_{2}(t, ω) of the first two terms in the Taylor series, centered at ω_{0}, of its numerator serves as the numerator for the particular solution x_{0}(t) of the homogeneous equation corresponding to (4). Therefore, the numerator in the difference x_{p}(t) − x_{0}(t) is the remainder

R_{2}(t, ω) = Ae^{ωt} − S_{2}(t, ω) = \frac{At^{2}}{2} e^{ω_{0}t}(ω − ω_{0})^{2} + ⋯

of the above Taylor series after the second term, and

x(t, ω) = \frac{R_{2}(t, ω)}{(ω − ω_{0})^{2}} .

Since the first term of R_{2}(t, ω) includes the factor (ω − ω_{0})^{2} and all other terms include higher powers of ω − ω_{0}, we conclude that the desired solution of (4) for ω = ω_{0} is simply the first term of R_{2}(t, ω) divided by (ω − ω_{0})^{2}.

A more general equation

(6)  L[x] = Ae^{ωt},

where L is a linear differential operator with constant coefficients, can be treated similarly. Assuming ω is not a characteristic root, that is L(ω) ≠ 0, we find by the method of undetermined coefficients a particular solution of (6),

(7)  x_{p}(t) = \frac{A}{L(ω)}e^{ωt}.

If ω_{0} is a characteristic root of order m, a particular solution of equation (6) cannot be obtained from (7) when ω = ω_{0}. To resolve the difficulty, we expand Ae^{ωt} in a Taylor series centered at ω_{0} and note that the partial sum of the first m terms,

S_{m}(t, ω) = Ae^{ω_{0}t} ∑_{i=0}^{m−1} \frac{t^{i}}{i!} (ω − ω_{0})^{i},

is a particular solution of the homogeneous equation L[x] = 0. If L(ω) ≠ 0, then

x_{0}(t) = \frac{S_{m}(t, ω)}{L(ω)}

is also a solution of the equation L[x] = 0 and

(8)  x(t, ω) = x_{p}(t) − x_{0}(t) = \frac{R_{m}(t, ω)}{L(ω)}

is a particular solution of (6), where

R_{m}(t, ω) = Ae^{ωt} − S_{m}(t, ω).

The limit of (8) as ω → ω_{0} represents a particular solution of (6) for the case ω = ω_{0}. Successively applying L’Hospital’s rule to (8) gives

(9) lim_{ω→ω_{0}} x(t, ω) = \frac{At^{m}}{L^{(m)}(ω_{0})}e^{ω_{0}t}.

This provides another explanation to the student why a particular solution of (6) must be sought in the form Bt^{m}e^{ωt} if ω is a zero of the characteristic polynomial. The reason why solution (8) “works” for all ω also becomes clear from the structure of its numerator and denominator. Indeed, since R_{m}(t, ω) is the remainder of the power series for Ae^{ωt} after the mth term and the polynomial L(ω) has a root of order m at ω_{0}, we can write

R_{m}(t, ω) = \frac{At^{m}}{m!} e^{ω_{0}t}(ω − ω_{0})^{m} + ⋯,

L(ω) = \frac{L^{(m)}(ω_{0})}{m!}(ω − ω_{0})^{m} + ⋯,

where the derivative L^{(m)}(ω_{0}) ≠ 0 and all missing terms contain powers of ω − ω_{0} higher than m. Substituting these expressions in (8) and approaching ω to ω_{0} yields (9).

The above results may be viewed as an application to differential equations of the powerful method of differentiation with respect to a parameter, which is important in the theory of improper integrals. This method can also be used to find particular solutions of linear differential equations in the “non–resonance” cases. Assume we want to find a particular solution of the equation

(10)  x′′ + 2x′ − 3x = t^{2}e^{5t}.

The usual way is to apply the method of undetermined coefficients but we can achieve the result also by differentiating twice a particular solution of the equation

(11)  y′′ + 2y′ − 3y = e^{ωt}

with respect to the parameter ω. The roots of the corresponding characteristic equation k^{2} + 2k − 3 = 0 are k = 1, −3. Therefore, to avoid the “resonance” case in (11), we take ω ≠ 1, −3. A particular solution of (11) is sought in the form y_{p}(t, ω) = Be^{ωt}, and substituting it in (11) gives

y_{p}(t, ω) = \frac{1}{ω^{2} + 2ω − 3}e^{ωt}

or

y_{p}(t, ω) = \frac{1}{4}[\frac{1}{ω − 1} − \frac{1}{ω + 3}]e^{ωt}.

Differentiating y_{p}(t, ω) twice with respect to ω provides a particular

solution x_{p}(t, ω) of the equation

x′′ + 2x′ − 3x = t^{2}e^{ωt},

that is,

x_{p}(t, ω) = \frac{1}{4}[\frac{1}{ω − 1} − \frac{1}{ω + 3}]t^{2}e^{ωt}

− \frac{1}{2}[\frac{1}{(ω − 1)^{2}} − \frac{1}{(ω + 3)^{2}} ]te^{ωt}

+ \frac{1}{2}[\frac{1}{(ω − 1)^{3}} − \frac{1}{(ω + 3)^{3}} ]e^{ωt}.

To obtain from here a particular solution of (10), it remains to put

ω = 5. Then

x_{p}(t) = \frac{32t^{2} − 24t + 7}{1024}e^{5t}.

Acknowledgement. I am thankful to the referee for a number of useful suggestions.

\underline{Reference}

1. R. Euler, “A Note on a Differential Equation,” Missouri Journal of Mathematical Sciences, Winter (1989), Vol. 1: No. 1, 26–27.