<< Chapter < Page Chapter >> Page >
y ( x ) = c 1 e ( α + β i ) x + c 2 e ( α β i ) x .

Using some smart choices for c 1 and c 2 , and a little bit of algebraic manipulation, we can find two linearly independent, real-value solutions to [link] and express our general solution in those terms.

We encountered exponential functions with complex exponents earlier. One of the key tools we used to express these exponential functions in terms of sines and cosines was Euler’s formula , which tells us that

e i θ = cos θ + i sin θ

for all real numbers θ .

Going back to the general solution, we have

y ( x ) = c 1 e ( α + β i ) x + c 2 e ( α β i ) x = c 1 e α x e β i x + c 2 e α x e β i x = e α x ( c 1 e β i x + c 2 e β i x ) .

Applying Euler’s formula together with the identities cos ( x ) = cos x and sin ( x ) = sin x , we get

y ( x ) = e α x [ c 1 ( cos β x + i sin β x ) + c 2 ( cos ( β x ) + i sin ( β x ) ) ] = e α x [ ( c 1 + c 2 ) cos β x + ( c 1 c 2 ) i sin β x ] .

Now, if we choose c 1 = c 2 = 1 2 , the second term is zero and we get

y ( x ) = e α x cos β x

as a real-value solution to [link] . Similarly, if we choose c 1 = i 2 and c 2 = i 2 , the first term is zero and we get

y ( x ) = e α x sin β x

as a second, linearly independent, real-value solution to [link] .

Based on this, we see that if the characteristic equation has complex conjugate roots α ± β i , then the general solution to [link] is given by

y ( x ) = c 1 e α x cos β x + c 2 e α x sin β x = e α x ( c 1 cos β x + c 2 sin β x ) ,

where c 1 and c 2 are constants.

For example, the differential equation y 2 y + 5 y = 0 has the associated characteristic equation λ 2 2 λ + 5 = 0 . By the quadratic formula, the roots of the characteristic equation are 1 ± 2 i . Therefore, the general solution to this differential equation is

y ( x ) = e x ( c 1 cos 2 x + c 2 sin 2 x ) .

Summary of results

We can solve second-order, linear, homogeneous differential equations with constant coefficients by finding the roots of the associated characteristic equation. The form of the general solution varies, depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots. The three cases are summarized in [link] .

Summary of characteristic equation cases
Characteristic Equation Roots General Solution to the Differential Equation
Distinct real roots, λ 1 and λ 2 y ( x ) = c 1 e λ 1 x + c 2 e λ 2 x
A repeated real root, λ y ( x ) = c 1 e λ x + c 2 x e λ x
Complex conjugate roots α ± β i y ( x ) = e α x ( c 1 cos β x + c 2 sin β x )

Problem-solving strategy: using the characteristic equation to solve second-order differential equations with constant coefficients

  1. Write the differential equation in the form a y + b y + c y = 0 .
  2. Find the corresponding characteristic equation a λ 2 + b λ + c = 0 .
  3. Either factor the characteristic equation or use the quadratic formula to find the roots.
  4. Determine the form of the general solution based on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots.

Solving second-order equations with constant coefficients

Find the general solution to the following differential equations. Give your answers as functions of x .

  1. y + 3 y 4 y = 0
  2. y + 6 y + 13 y = 0
  3. y + 2 y + y = 0
  4. y 5 y = 0
  5. y 16 y = 0
  6. y + 16 y = 0

Note that all these equations are already given in standard form (step 1).

  1. The characteristic equation is λ 2 + 3 λ 4 = 0 (step 2). This factors into ( λ + 4 ) ( λ 1 ) = 0 , so the roots of the characteristic equation are λ 1 = −4 and λ 2 = 1 (step 3). Then the general solution to the differential equation is
    y ( x ) = c 1 e −4 x + c 2 e x (step 4) .
  2. The characteristic equation is λ 2 + 6 λ + 13 = 0 (step 2). Applying the quadratic formula, we see this equation has complex conjugate roots −3 ± 2 i (step 3). Then the general solution to the differential equation is
    y ( t ) = e −3 t ( c 1 cos 2 t + c 2 sin 2 t ) (step 4) .
  3. The characteristic equation is λ 2 + 2 λ + 1 = 0 (step 2). This factors into ( λ + 1 ) 2 = 0 , so the characteristic equation has a repeated real root λ = −1 (step 3). Then the general solution to the differential equation is
    y ( t ) = c 1 e t + c 2 t e t (step 4).
  4. The characteristic equation is λ 2 5 λ (step 2). This factors into λ ( λ 5 ) = 0 , so the roots of the characteristic equation are λ 1 = 0 and λ 2 = 5 (step 3). Note that e 0 x = e 0 = 1 , so our first solution is just a constant. Then the general solution to the differential equation is
    y ( x ) = c 1 + c 2 e 5 x (step 4) .
  5. The characteristic equation is λ 2 16 = 0 (step 2). This factors into ( λ + 4 ) ( λ 4 ) = 0 , so the roots of the characteristic equation are λ 1 = 4 and λ 2 = −4 (step 3). Then the general solution to the differential equation is
    y ( x ) = c 1 e 4 x + c 2 e −4 x (step 4) .
  6. The characteristic equation is λ 2 + 16 = 0 (step 2). This has complex conjugate roots ± 4 i (step 3). Note that e 0 x = e 0 = 1 , so the exponential term in our solution is just a constant. Then the general solution to the differential equation is
    y ( t ) = c 1 cos 4 t + c 2 sin 4 t (step 4) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask