TMA4110 - Calculus 3 - Lecture 6 · Consider the second-order homogeneous linear differential equation y00+py0+qy = 0 with constant coefﬁcients. The characteristic polynomial of

TMA4110 - Calculus 3Lecture 6

Toke Meier CarlsenNorwegian University of Science and Technology

Fall 2012

www.ntnu.no TMA4110 - Calculus 3, Lecture 6

Reference group

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Onsager Lecture

The Lars Onsager Lecture for 2012 will be presented by ProfessorArnold J. Levine, Institute for Advance Study, Princeton. Monday,September 10 at 11:15–12:00 in S2 he will present the LarsOnsager Lecture 2012 with title Searching for the Origins ofCancers so as to Inform Treatments.See http://www.ntnu.edu/onsager for more information.


Second-order homogeneous lineardifferential equations with constantcoefficients

Consider the second-order homogeneous linear differentialequation

y ′′ + py ′ + qy = 0

with constant coefficients.

The characteristic polynomial of the equation is the polynomialλ2 + pλ+ q.The roots

λ =−p ±

√p2 − 4q

2of λ2 + pλ+ q are called thecharacteristic roots of the equation.




y ′′ + py ′ + qy = 0

with constant coefficients.

The characteristic polynomial of the equation is the polynomialλ2 + pλ+ q.The roots

λ =−p ±

√p2 − 4q





y ′′ + py ′ + qy = 0

with constant coefficients.The characteristic polynomial of the equation is the polynomialλ2 + pλ+ q.

The roots

λ =−p ±

√p2 − 4q





y ′′ + py ′ + qy = 0

with constant coefficients.The characteristic polynomial of the equation is the polynomialλ2 + pλ+ q.The roots

λ =−p ±

√p2 − 4q



General solutions to second-orderhomogeneous linear differentialequations with constant coefficients

If p2 − 4q > 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct real roots λ1 and λ2, and the general solutionof y ′′ + py ′ + qy = 0 is c1eλ1t + c2eλ2t .If p2 − 4q < 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct complex roots λ1 = a + ib and λ2 = a− ib,and the general solution of y ′′ + py ′ + qy = 0 isc1eat cos(bt) + c2eat sin(bt).If p2 − 4q = 0, then the characteristic polynomial λ2 + pλ+ qjust have one root λ, and the general solution ofy ′′ + py ′ + qy = 0 is c1eλt + c2teλt .



If p2 − 4q > 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct real roots λ1 and λ2, and the general solutionof y ′′ + py ′ + qy = 0 is c1eλ1t + c2eλ2t .

If p2 − 4q < 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct complex roots λ1 = a + ib and λ2 = a− ib,and the general solution of y ′′ + py ′ + qy = 0 isc1eat cos(bt) + c2eat sin(bt).If p2 − 4q = 0, then the characteristic polynomial λ2 + pλ+ qjust have one root λ, and the general solution ofy ′′ + py ′ + qy = 0 is c1eλt + c2teλt .



If p2 − 4q > 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct real roots λ1 and λ2, and the general solutionof y ′′ + py ′ + qy = 0 is c1eλ1t + c2eλ2t .If p2 − 4q < 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct complex roots λ1 = a + ib and λ2 = a− ib,and the general solution of y ′′ + py ′ + qy = 0 isc1eat cos(bt) + c2eat sin(bt).

If p2 − 4q = 0, then the characteristic polynomial λ2 + pλ+ qjust have one root λ, and the general solution ofy ′′ + py ′ + qy = 0 is c1eλt + c2teλt .



If p2 − 4q > 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct real roots λ1 and λ2, and the general solutionof y ′′ + py ′ + qy = 0 is c1eλ1t + c2eλ2t .If p2 − 4q < 0, then the characteristic polynomial λ2 + pλ+ qhas two distinct complex roots λ1 = a + ib and λ2 = a− ib,and the general solution of y ′′ + py ′ + qy = 0 isc1eat cos(bt) + c2eat sin(bt).If p2 − 4q = 0, then the characteristic polynomial λ2 + pλ+ qjust have one root λ, and the general solution ofy ′′ + py ′ + qy = 0 is c1eλt + c2teλt .


Harmonic motion

x = 0

x = x0

We consider a spring suspendedfrom a beam.

The position of thebottom of the spring is thereference point from which wemeasure displacement, so itcorresponds to x = 0.We then attach a weight of massm to the spring. This weightstretches the spring until it isonce more in equilibrium atx = x0.


Harmonic motion

x = 0

x = x0

We consider a spring suspendedfrom a beam. The position of thebottom of the spring is thereference point from which wemeasure displacement, so itcorresponds to x = 0.

We then attach a weight of massm to the spring. This weightstretches the spring until it isonce more in equilibrium atx = x0.


Harmonic motion

x = 0

x = x0

We consider a spring suspendedfrom a beam. The position of thebottom of the spring is thereference point from which wemeasure displacement, so itcorresponds to x = 0.We then attach a weight of massm to the spring. This weightstretches the spring until it isonce more in equilibrium atx = x0.


Harmonic motion

x = 0

x = x0

At this point there are two forcesacting on the mass. There is theforce of gravity mg, and there isthe restoring force of the springwhich we denote by R(x) since itdepends on the distance x thatthe spring is stretched.

Since we have equilibrium atx = x0, the total force on theweight is 0, so R(x0) + mg = 0.


Harmonic motion

x = 0

x = x0

At this point there are two forcesacting on the mass. There is theforce of gravity mg, and there isthe restoring force of the springwhich we denote by R(x) since itdepends on the distance x thatthe spring is stretched.Since we have equilibrium atx = x0, the total force on theweight is 0, so R(x0) + mg = 0.


Harmonic motion

x = 0

x = x0

We now set the mass in motionby stretching the spring further.

In addition to gravity and therestoring force, there is adamping force D which is theresistance to the motion of theweight due to the mediumthrough which the weight ismoving and perhaps tosomething internal to the spring.We assume that D depends onthe velocity x ′ of the mass, andwrite it as D(x ′).


Harmonic motion

x = 0

x = x0

We now set the mass in motionby stretching the spring further.In addition to gravity and therestoring force, there is adamping force D which is theresistance to the motion of theweight due to the mediumthrough which the weight ismoving and perhaps tosomething internal to the spring.

We assume that D depends onthe velocity x ′ of the mass, andwrite it as D(x ′).


Harmonic motion

x = 0

x = x0

We now set the mass in motionby stretching the spring further.In addition to gravity and therestoring force, there is adamping force D which is theresistance to the motion of theweight due to the mediumthrough which the weight ismoving and perhaps tosomething internal to the spring.We assume that D depends onthe velocity x ′ of the mass, andwrite it as D(x ′).


Harmonic motion

x = 0

x = x0

According to Newton’s secondlaw, we have

mx ′′ = R(x) + mg + D(x ′).

We assume that R(x) = −kx forsome positive constant k calledthe spring constant, and thatD(x ′) = −µx ′ for somenonnegative constant µ calledthe damping constant.Thus we havemx ′′ = −kx + mg − µx ′.


Harmonic motion

x = 0

x = x0


mx ′′ = R(x) + mg + D(x ′).

We assume that R(x) = −kx forsome positive constant k calledthe spring constant, and thatD(x ′) = −µx ′ for somenonnegative constant µ calledthe damping constant.

Thus we havemx ′′ = −kx + mg − µx ′.


Harmonic motion

x = 0

x = x0


mx ′′ = R(x) + mg + D(x ′).

We assume that R(x) = −kx forsome positive constant k calledthe spring constant, and thatD(x ′) = −µx ′ for somenonnegative constant µ calledthe damping constant.Thus we havemx ′′ = −kx + mg − µx ′.


Harmonic motion

x = 0

x = x0

Recall that R(x0) + mg = 0.

Somg = −R(x0) = kx0. If we lety = x − x0, thenmx ′′ = −kx + mg − µx ′ becomesmy ′′ + µy ′ + ky = 0 becausex ′ = y ′ and x ′′ = y ′′.If we let ω0 =

√k/m and

c = µ/2m, then the aboveequation becomes

y ′′ + 2cy ′ + ω20y = 0

where c ≥ 0 and ω0 > 0.


Harmonic motion

x = 0

x = x0

Recall that R(x0) + mg = 0. Somg = −R(x0) = kx0.

If we lety = x − x0, thenmx ′′ = −kx + mg − µx ′ becomesmy ′′ + µy ′ + ky = 0 becausex ′ = y ′ and x ′′ = y ′′.If we let ω0 =

√k/m and


y ′′ + 2cy ′ + ω20y = 0



Harmonic motion

x = 0

x = x0

Recall that R(x0) + mg = 0. Somg = −R(x0) = kx0. If we lety = x − x0, thenmx ′′ = −kx + mg − µx ′ becomesmy ′′ + µy ′ + ky = 0 becausex ′ = y ′ and x ′′ = y ′′.

If we let ω0 =√

k/m andc = µ/2m, then the aboveequation becomes

y ′′ + 2cy ′ + ω20y = 0



Harmonic motion

x = 0

x = x0

Recall that R(x0) + mg = 0. Somg = −R(x0) = kx0. If we lety = x − x0, thenmx ′′ = −kx + mg − µx ′ becomesmy ′′ + µy ′ + ky = 0 becausex ′ = y ′ and x ′′ = y ′′.If we let ω0 =

√k/m and


y ′′ + 2cy ′ + ω20y = 0



Harmonic motion

The motion described by a solution to the equation

y ′′ + 2cy ′ + ω20y = 0

where c ≥ 0 and ω0 > 0, is called a harmonic motion.


Simple harmonic motion

If c = 0 we say that the system is undamped.

In that case, theequation becomes

y ′′ + ω20y = 0

where ω0 > 0.The general solution to this equation is

y(t) = c1 cos(ω0t) + c2 sin(ω0t).

The motion described by this solution is called a simple harmonicmotion. The number ω0 =

√k/m is called the natural frequency.

The number T = 2π/ω0 = 2π/√

k/m is called the period.



If c = 0 we say that the system is undamped. In that case, theequation becomes

y ′′ + ω20y = 0

where ω0 > 0.

The general solution to this equation is









y ′′ + ω20y = 0










y ′′ + ω20y = 0



The motion described by this solution is called a simple harmonicmotion.

The number ω0 =√

k/m is called the natural frequency.The number T = 2π/ω0 = 2π/

√k/m is called the period.




y ′′ + ω20y = 0










y ′′ + ω20y = 0








Amplitude and phase angle

It is frequently convenient to put the solutiony(t) = c1 cos(ω0t) + c2 sin(ω0t) into another form that is moreconvenient and more revealing of the nature of the solution.

Let z = c1 + ic2. If we let A = |z| and φ = Arg(z), thenz = A(cos(φ) + i sin(φ)) from which it follows that c1 = A cos(φ)and c2 = A sin(φ), and that

y(t) = c1 cos(ω0t) + c2 sin(ω0t)= A cos(φ) cos(ω0t) + A sin(φ) sin(ω0t) = A cos(ω0t − φ).

The number A is called the amplitude, and the number φ is calledthe phase.



It is frequently convenient to put the solutiony(t) = c1 cos(ω0t) + c2 sin(ω0t) into another form that is moreconvenient and more revealing of the nature of the solution.Let z = c1 + ic2.

If we let A = |z| and φ = Arg(z), thenz = A(cos(φ) + i sin(φ)) from which it follows that c1 = A cos(φ)and c2 = A sin(φ), and that





It is frequently convenient to put the solutiony(t) = c1 cos(ω0t) + c2 sin(ω0t) into another form that is moreconvenient and more revealing of the nature of the solution.Let z = c1 + ic2. If we let A = |z| and φ = Arg(z), thenz = A(cos(φ) + i sin(φ)) from which it follows that c1 = A cos(φ)and c2 = A sin(φ),

and that





It is frequently convenient to put the solutiony(t) = c1 cos(ω0t) + c2 sin(ω0t) into another form that is moreconvenient and more revealing of the nature of the solution.Let z = c1 + ic2. If we let A = |z| and φ = Arg(z), thenz = A(cos(φ) + i sin(φ)) from which it follows that c1 = A cos(φ)and c2 = A sin(φ), and that





It is frequently convenient to put the solutiony(t) = c1 cos(ω0t) + c2 sin(ω0t) into another form that is moreconvenient and more revealing of the nature of the solution.Let z = c1 + ic2. If we let A = |z| and φ = Arg(z), thenz = A(cos(φ) + i sin(φ)) from which it follows that c1 = A cos(φ)and c2 = A sin(φ), and that





t

y

y(t) = A cos(ω0t − φ)

φ/ω0

T = 2π/ω0

A


The underdamped case

If 0 < c < ω0, then the characteristic roots of y ′′ + 2cy ′ + ω20y = 0

are

λ =−2c ±

√(2c)2 − 4ω0

2= −c ±

√c2 − ω2

0 = −c ± i√ω2

0 − c2

so the the general solution to y ′′ + 2cy ′ + ω20y = 0 is

y(t) = e−ct(c1 cos(ωt) + c2 sin(ωt))

where ω =√ω2

0 − c2.




are

λ =−2c ±

√(2c)2 − 4ω0

2= −c ±

√c2 − ω2

0 = −c ± i√ω2

0 − c2



where ω =√ω2

0 − c2.




are

λ =−2c ±

√(2c)2 − 4ω0

2= −c ±

√c2 − ω2

0 = −c ± i√ω2

0 − c2



where ω =√ω2

0 − c2.



t

y



The overdamped case

If c > ω0, then the characteristic roots of y ′′ + 2cy ′ + ω20y = 0 are

λ =−2c ±

√(2c)2 − 4ω0

2= −c ±

√c2 − ω2

0


y(t) = c1eλ1t + c2eλ2t

where λ1 = −c −√

c2 − ω20 and λ2 = −c +

√c2 − ω2

0.


The overdamped case


λ =−2c ±

√(2c)2 − 4ω0

2= −c ±

√c2 − ω2

0




c2 − ω20 and λ2 = −c +

√c2 − ω2

0.


The overdamped case


λ =−2c ±

√(2c)2 − 4ω0

2= −c ±

√c2 − ω2

0




c2 − ω20 and λ2 = −c +

√c2 − ω2

0.


The overdamped case

t

y



The critically damped case

If c = ω0, then the equation y ′′ + 2cy ′ + ω20y = 0 only has one

characteristic root

λ =−2c ±

√(2c)2 − 4ω0

2= −c


y(t) = c1e−ct + c2te−ct .




characteristic root

λ =−2c ±

√(2c)2 − 4ω0

2= −c






characteristic root

λ =−2c ±

√(2c)2 − 4ω0

2= −c





t

y

y(t) = c1e−ct + c2te−ct


Inhomogeneous equations

We now turn to the solution of inhomogeneous second-order lineardifferential equations

y ′′ + py ′ + qy = f

where p = p(t), q = q(t) and f = f (t) are functions of theindependent variable.Suppose we have found a particular solution yp to the equation.If yh is a solution to the homogeneous equation y ′′ + py ′ + qy = 0,then yp + yh is a solution to the inhomogeneous equationy ′′ + py ′ + qy = f because (yp + yh)

′′ + p(yp + yh)′ + q(yp + yh) =

(y ′′p + py ′p + qyp) + (y ′′h + py ′h + qyh) = f + 0 = f .




y ′′ + py ′ + qy = f

where p = p(t), q = q(t) and f = f (t) are functions of theindependent variable.

Suppose we have found a particular solution yp to the equation.If yh is a solution to the homogeneous equation y ′′ + py ′ + qy = 0,then yp + yh is a solution to the inhomogeneous equationy ′′ + py ′ + qy = f because (yp + yh)

′′ + p(yp + yh)′ + q(yp + yh) =





y ′′ + py ′ + qy = f

where p = p(t), q = q(t) and f = f (t) are functions of theindependent variable.Suppose we have found a particular solution yp to the equation.

If yh is a solution to the homogeneous equation y ′′ + py ′ + qy = 0,then yp + yh is a solution to the inhomogeneous equationy ′′ + py ′ + qy = f because (yp + yh)

′′ + p(yp + yh)′ + q(yp + yh) =





y ′′ + py ′ + qy = f

where p = p(t), q = q(t) and f = f (t) are functions of theindependent variable.Suppose we have found a particular solution yp to the equation.If yh is a solution to the homogeneous equation y ′′ + py ′ + qy = 0,then yp + yh is a solution to the inhomogeneous equationy ′′ + py ′ + qy = f

because (yp + yh)′′ + p(yp + yh)

′ + q(yp + yh) =(y ′′p + py ′p + qyp) + (y ′′h + py ′h + qyh) = f + 0 = f .




y ′′ + py ′ + qy = f

where p = p(t), q = q(t) and f = f (t) are functions of theindependent variable.Suppose we have found a particular solution yp to the equation.If yh is a solution to the homogeneous equation y ′′ + py ′ + qy = 0,then yp + yh is a solution to the inhomogeneous equationy ′′ + py ′ + qy = f because (yp + yh)

′′ + p(yp + yh)′ + q(yp + yh) =




Conversely, if yp1 and yp2 are two different solutions to theinhomogeneous equation y ′′ + py ′ + qy = f , then yh = yp1 − yp2 isa solution to the homogeneous equation y ′′ + py ′ + qy = 0

because (yp1 − yp2)′′ + p(yp1 − yp2)

′ + q(yp1 − yp2) =(y ′′p1

+ py ′p1+ qyp1)− (y ′′p2

+ py ′p2+ qyp2) = f − f = 0.



Conversely, if yp1 and yp2 are two different solutions to theinhomogeneous equation y ′′ + py ′ + qy = f , then yh = yp1 − yp2 isa solution to the homogeneous equation y ′′ + py ′ + qy = 0because (yp1 − yp2)

′′ + p(yp1 − yp2)′ + q(yp1 − yp2) =

(y ′′p1+ py ′p1

+ qyp1)− (y ′′p2+ py ′p2

+ qyp2) = f − f = 0.



It follows that if yp is a particular solution to the inhomogeneousequation y ′′ + py ′ + qy = f and y1 and y2 form a fundamental set ofsolutions to the homogeneous equation y ′′ + py ′ + qy = 0, then thegeneral solution to the inhomogeneous equation y ′′ + py ′ + qy = fis

y = yp + c1y1 + c2y2

where c1 and c2 are constants.


The method of undetermined coefficients

Consider the inhomogeneous second-order linear differentialequation

y ′′ + py ′ + qy = f .

If the function f has a form that is replicated under differentiation,then look for a solution with the same general form as f .




y ′′ + py ′ + qy = f .





y ′′ + py ′ + qy = f .



Exponential forcing terms

If f (t) = eat , then f ′(t) = aeat , so we will look for a solution of theform beat .


Exponential forcing terms

If f (t) = eat , then f ′(t) = aeat , so we will look for a solution of theform beat .


Example

Let us find a particular solution to the equation

y ′′ − y ′ − 2y = 2e−2t .


Example


y ′′ − y ′ − 2y = 2e−2t .


Example


y ′′ − y ′ − 2y = 2e−2t .


Example


y ′′ − y ′ − 2y = 2e−2t .


Example


y ′′ − y ′ − 2y = 2e−2t .


Trigonometric forcing terms

If f (t) = A cos(ωt) + B sin(ωt), thenf ′(t) = −ωA sin(ωt) + ωB cos(ωt), so we will look for a solution ofthe form a cos(ωt) + b sin(ωt).


Trigonometric forcing terms

If f (t) = A cos(ωt) + B sin(ωt), thenf ′(t) = −ωA sin(ωt) + ωB cos(ωt), so we will look for a solution ofthe form a cos(ωt) + b sin(ωt).


Example


y ′′ + 2y ′ − 3y = 5 sin(3t).


Example


y ′′ + 2y ′ − 3y = 5 sin(3t).


Example


y ′′ + 2y ′ − 3y = 5 sin(3t).


Example


y ′′ + 2y ′ − 3y = 5 sin(3t).


Example


y ′′ + 2y ′ − 3y = 5 sin(3t).


Example


y ′′ + 2y ′ − 3y = 5 sin(3t).


The complex method

Consider again the equation

y ′′ + 2y ′ − 3y = 5 sin(3t).

Since 5 sin(3t) = Im(5e3it), y(t) = Im(z(t)) is a solution toy ′′ + 2y ′ − 3y = 5 sin(3t) if z is a solution to z ′′ + 2z ′ − 3z = 5e3it .


The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).

Since 5 sin(3t) = Im(5e3it),

y(t) = Im(z(t)) is a solution toy ′′ + 2y ′ − 3y = 5 sin(3t) if z is a solution to z ′′ + 2z ′ − 3z = 5e3it .


The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



The complex method


y ′′ + 2y ′ − 3y = 5 sin(3t).



Polynomial forcing terms

If f (t) = antn + an−1tn−1 + . . . a1t + a0, thenf ′(t) = nantn−1 + (n − 1)an−1tn−2 + . . . a1, so we will look for asolution of the form bntn + bn−1tn−1 + . . . b1t + b0.


Polynomial forcing terms

If f (t) = antn + an−1tn−1 + . . . a1t + a0, thenf ′(t) = nantn−1 + (n − 1)an−1tn−2 + . . . a1, so we will look for asolution of the form bntn + bn−1tn−1 + . . . b1t + b0.


Example


y ′′ + 2y ′ − 3y = 3t + 4.


Example


y ′′ + 2y ′ − 3y = 3t + 4.


Example


y ′′ + 2y ′ − 3y = 3t + 4.


Example


y ′′ + 2y ′ − 3y = 3t + 4.


Example


y ′′ + 2y ′ − 3y = 3t + 4.


Exceptional cases

The method of undetermined coefficients looks straightforward.There are, however, some exceptional cases to look out for. If theforcing term f , and hence the proposed solution, is a solution to thehomogeneous equation y ′′ + py ′ + qy = 0, then the proposedsolution wouldn’t work.

Instead we have to multiply the proposed solution by t .


Exceptional cases

The method of undetermined coefficients looks straightforward.There are, however, some exceptional cases to look out for. If theforcing term f , and hence the proposed solution, is a solution to thehomogeneous equation y ′′ + py ′ + qy = 0, then the proposedsolution wouldn’t work.Instead we have to multiply the proposed solution by t .


Example


y ′′ − y ′ − 2y = 3e−t .


Example


y ′′ − y ′ − 2y = 3e−t .


Example


y ′′ − y ′ − 2y = 3e−t .


Example


y ′′ − y ′ − 2y = 3e−t .


Example


y ′′ − y ′ − 2y = 3e−t .


Example


y ′′ − y ′ − 2y = 3e−t .


Example


y ′′ − y ′ − 2y = 3e−t .


Combination forcing terms

If yf is a solution the differential equation y ′′ + py ′ + qy = f , yg is asolution the differential equation y ′′ + py ′ + qy = g, and c1 and c2are constants, then y(t) = c1yf (t) + c2yg(t) is a solution to thedifferential equation y ′′ + py ′ + qy = c1f + c2g.


Example


y ′′ − y ′ − 2y = e−2t − 3e−t .


Example


y ′′ − y ′ − 2y = e−2t − 3e−t .


Example


y ′′ − y ′ − 2y = e−2t − 3e−t .


Example

Let us find the general solution to the equation

y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Example


y ′′ + 4y ′ + 4y = 4− t .


Documents

TMA4110 - Calculus 3 - Lecture 6 · Consider the second-order homogeneous linear differential equation y00+py0+qy = 0 with constant coefﬁcients. The characteristic polynomial of