# 7.2 Nonhomogeneous linear equations  (Page 2/9)

 Page 2 / 9

## Undetermined coefficients

The method of undetermined coefficients    involves making educated guesses about the form of the particular solution based on the form of $r\left(x\text{).}$ When we take derivatives of polynomials, exponential functions, sines, and cosines, we get polynomials, exponential functions, sines, and cosines. So when $r\left(x\right)$ has one of these forms, it is possible that the solution to the nonhomogeneous differential equation might take that same form. Let’s look at some examples to see how this works.

## Undetermined coefficients when $r\left(x\right)$ Is a polynomial

Find the general solution to $y\text{″}+4{y}^{\prime }+3y=3x\text{.}$

The complementary equation is $y\text{″}+4{y}^{\prime }+3y=0,$ with general solution ${c}_{1}{e}^{\text{−}x}+{c}_{2}{e}^{-3x}.$ Since $r\left(x\right)=3x,$ the particular solution might have the form ${y}_{p}\left(x\right)=Ax+B\text{.}$ If this is the case, then we have ${y}_{p}{}^{\prime }\left(x\right)=A$ and ${y}_{p}\text{″}\left(x\right)=0.$ For ${y}_{p}$ to be a solution to the differential equation, we must find values for $A$ and $B$ such that

$\begin{array}{ccc}\hfill y\text{″}+4{y}^{\prime }+3y& =\hfill & 3x\hfill \\ \hfill 0+4\left(A\right)+3\left(Ax+B\right)& =\hfill & 3x\hfill \\ \hfill 3Ax+\left(4A+3B\right)& =\hfill & 3x\text{.}\hfill \end{array}$

Setting coefficients of like terms equal, we have

$\begin{array}{ccc}\hfill 3A& =\hfill & 3\hfill \\ \hfill 4A+3B& =\hfill & 0.\hfill \end{array}$

Then, $A=1$ and $B=-\frac{4}{3},$ so ${y}_{p}\left(x\right)=x-\frac{4}{3}$ and the general solution is

$y\left(x\right)={c}_{1}{e}^{\text{−}x}+{c}_{2}{e}^{-3x}+x-\frac{4}{3}.$

In [link] , notice that even though $r\left(x\right)$ did not include a constant term, it was necessary for us to include the constant term in our guess. If we had assumed a solution of the form ${y}_{p}=Ax$ (with no constant term), we would not have been able to find a solution. (Verify this!) If the function $r\left(x\right)$ is a polynomial, our guess for the particular solution should be a polynomial of the same degree, and it must include all lower-order terms, regardless of whether they are present in $r\left(x\text{).}$

## Undetermined coefficients when $r\left(x\right)$ Is an exponential

Find the general solution to $y\text{″}-{y}^{\prime }-2y=2{e}^{3x}.$

The complementary equation is $y\text{″}-{y}^{\prime }-2y=0,$ with the general solution ${c}_{1}{e}^{\text{−}x}+{c}_{2}{e}^{2x}.$ Since $r\left(x\right)=2{e}^{3x},$ the particular solution might have the form ${y}_{p}\left(x\right)=A{e}^{3x}.$ Then, we have ${y}_{p}{}^{\prime }\left(x\right)=3A{e}^{3x}$ and ${y}_{p}\text{″}\left(x\right)=9A{e}^{3x}.$ For ${y}_{p}$ to be a solution to the differential equation, we must find a value for $A$ such that

$\begin{array}{ccc}\hfill y\text{″}-{y}^{\prime }-2y& =\hfill & 2{e}^{3x}\hfill \\ \hfill 9A{e}^{3x}-3A{e}^{3x}-2A{e}^{3x}& =\hfill & 2{e}^{3x}\hfill \\ \hfill 4A{e}^{3x}& =\hfill & 2{e}^{3x}.\hfill \end{array}$

So, $4A=2$ and $A=1\text{/}2.$ Then, ${y}_{p}\left(x\right)=\left(\frac{1}{2}\right){e}^{3x},$ and the general solution is

$y\left(x\right)={c}_{1}{e}^{\text{−}x}+{c}_{2}{e}^{2x}+\frac{1}{2}{e}^{3x}.$

Find the general solution to $y\text{″}-4{y}^{\prime }+4y=7\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t-\text{cos}\phantom{\rule{0.1em}{0ex}}t\text{.}$

$y\left(t\right)={c}_{1}{e}^{2t}+{c}_{2}t{e}^{2t}+\text{sin}\phantom{\rule{0.1em}{0ex}}t+\text{cos}\phantom{\rule{0.1em}{0ex}}t$

In the previous checkpoint, $r\left(x\right)$ included both sine and cosine terms. However, even if $r\left(x\right)$ included a sine term only or a cosine term only, both terms must be present in the guess. The method of undetermined coefficients also works with products of polynomials, exponentials, sines, and cosines. Some of the key forms of $r\left(x\right)$ and the associated guesses for ${y}_{p}\left(x\right)$ are summarized in [link] .

Key forms for the method of undetermined coefficients
$r\left(x\right)$ Initial guess for ${y}_{p}\left(x\right)$
$k$ (a constant) $A$ (a constant)
$ax+b$ $Ax+B$ ( Note : The guess must include both terms even if $b=0.$ )
$a{x}^{2}+bx+c$ $A{x}^{2}+Bx+C$ ( Note : The guess must include all three terms even if $b$ or $c$ are zero.)
Higher-order polynomials Polynomial of the same order as $r\left(x\right)$
$a{e}^{\lambda x}$ $A{e}^{\lambda x}$
$a\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x+b\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x$ $A\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x+B\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x$ ( Note : The guess must include both terms even if either $a=0$ or $b=0.$ )
$a{e}^{\alpha x}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x+b{e}^{\alpha x}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x$ $A{e}^{\alpha x}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x+B{e}^{\alpha x}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x$
$\left(a{x}^{2}+bx+c\right){e}^{\lambda x}$ $\left(A{x}^{2}+Bx+C\right){e}^{\lambda x}$
$\begin{array}{l}\left({a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \\ +\left({b}_{2}{x}^{2}+{b}_{1}x+{b}_{0}\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \end{array}$ $\begin{array}{l}\left({A}_{2}{x}^{2}+{A}_{1}x+{A}_{0}\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \\ +\left({B}_{2}{x}^{2}+{B}_{1}x+{B}_{0}\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \end{array}$
$\begin{array}{l}\left({a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}\right){e}^{\alpha x}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \\ +\left({b}_{2}{x}^{2}+{b}_{1}x+{b}_{0}\right){e}^{\alpha x}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \end{array}$ $\begin{array}{l}\left({A}_{2}{x}^{2}+{A}_{1}x+{A}_{0}\right){e}^{\alpha x}\text{cos}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \\ +\left({B}_{2}{x}^{2}+{B}_{1}x+{B}_{0}\right){e}^{\alpha x}\text{sin}\phantom{\rule{0.1em}{0ex}}\beta x\hfill \end{array}$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!