# 4.5 First-order linear equations  (Page 6/10)

 Page 6 / 10

A circuit has in series an electromotive force given by $E=20\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}5t$ V, a capacitor with capacitance $0.02\phantom{\rule{0.2em}{0ex}}\text{F},$ and a resistor of $8\phantom{\rule{0.2em}{0ex}}\text{Ω}.$ If the initial charge is $4\phantom{\rule{0.2em}{0ex}}\text{C},$ find the charge at time $t>0.$

Initial-value problem:

$8{q}^{\prime }+\frac{1}{0.02}q=20\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}5t,\phantom{\rule{1em}{0ex}}q\left(0\right)=4$

$q\left(t\right)=\frac{10\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}5t-8\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}5t+172{e}^{-6.25t}}{41}$

## Key concepts

• Any first-order linear differential equation can be written in the form $y\prime +p\left(x\right)y=q\left(x\right).$
• We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.
• Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.

## Key equations

• standard form
$y\prime +p\left(x\right)y=q\left(x\right)$
• integrating factor
$\mu \left(x\right)={e}^{\int p\left(x\right)\phantom{\rule{0.1em}{0ex}}dx}$

Are the following differential equations linear? Explain your reasoning.

$\frac{dy}{dx}={x}^{2}y+\text{sin}\phantom{\rule{0.1em}{0ex}}x$

$\frac{dy}{dt}=ty$

Yes

$\frac{dy}{dt}+{y}^{2}=x$

$y\prime ={x}^{3}+{e}^{x}$

Yes

$y\prime =y+{e}^{y}$

Write the following first-order differential equations in standard form.

$y\prime ={x}^{3}y+\text{sin}\phantom{\rule{0.1em}{0ex}}x$

$y\prime -{x}^{3}y=\text{sin}\phantom{\rule{0.1em}{0ex}}x$

$y\prime +3y-\text{ln}\phantom{\rule{0.1em}{0ex}}x=0$

$\text{−}xy\prime =\left(3x+2\right)y+x{e}^{x}$

$y\prime +\frac{\left(3x+2\right)}{x}y=\text{−}{e}^{x}$

$\frac{dy}{dt}=4y+ty+\text{tan}\phantom{\rule{0.1em}{0ex}}t$

$\frac{dy}{dt}=yx\left(x+1\right)$

$\frac{dy}{dt}-yx\left(x+1\right)=0$

What are the integrating factors for the following differential equations?

$y\prime =xy+3$

$y\prime +{e}^{x}y=\text{sin}\phantom{\rule{0.1em}{0ex}}x$

${e}^{x}$

$y\prime =x\phantom{\rule{0.1em}{0ex}}\text{ln}\left(x\right)y+3x$

$\frac{dy}{dx}=\text{tanh}\left(x\right)y+1$

$\text{−}\text{ln}\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}x\right)$

$\frac{dy}{dt}+3ty={e}^{t}y$

Solve the following differential equations by using integrating factors.

$y\prime =3y+2$

$y=C{e}^{3x}-\frac{2}{3}$

$y\prime =2y-{x}^{2}$

$xy\prime =3y-6{x}^{2}$

$y=C{x}^{3}+6{x}^{2}$

$\left(x+2\right)y\prime =3x+y$

$y\prime =3x+xy$

$y=C{e}^{{x}^{2}\text{/}2}-3$

$xy\prime =x+y$

$\text{sin}\left(x\right)y\prime =y+2x$

$y=C\phantom{\rule{0.1em}{0ex}}\text{tan}\left(\frac{x}{2}\right)-2x+4\phantom{\rule{0.1em}{0ex}}\text{tan}\left(\frac{x}{2}\right)\text{ln}\left(\text{sin}\left(\frac{x}{2}\right)\right)$

$y\prime =y+{e}^{x}$

$xy\prime =3y+{x}^{2}$

$y=C{x}^{3}-{x}^{2}$

$y\prime +\text{ln}\phantom{\rule{0.1em}{0ex}}x=\frac{y}{x}$

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

[T] $\left(x+2\right)y\prime =2y-1$

$y=C{\left(x+2\right)}^{2}+\frac{1}{2}$

[T] $y\prime =3{e}^{t\text{/}3}-2y$

[T] $xy\prime +\frac{y}{2}=\text{sin}\left(3t\right)$

$y=\frac{C}{\sqrt{x}}+2\phantom{\rule{0.1em}{0ex}}\text{sin}\left(3t\right)$

[T] $xy\prime =2\frac{\text{cos}\phantom{\rule{0.1em}{0ex}}x}{x}-3y$

[T] $\left(x+1\right)y\prime =3y+{x}^{2}+2x+1$

$y=C{\left(x+1\right)}^{3}-{x}^{2}-2x-1$

[T] $\text{sin}\left(x\right)y\prime +\text{cos}\left(x\right)y=2x$

[T] $\sqrt{{x}^{2}+1}y\prime =y+2$

$y=C{e}^{{\text{sinh}}^{-1}x}-2$

[T] ${x}^{3}y\prime +2{x}^{2}y=x+1$

Solve the following initial-value problems by using integrating factors.

$y\prime +y=x,y\left(0\right)=3$

$y=x+4{e}^{x}-1$

$y\prime =y+2{x}^{2},y\left(0\right)=0$

$xy\prime =y-3{x}^{3},y\left(1\right)=0$

$y=-\frac{3x}{2}\left({x}^{2}-1\right)$

${x}^{2}y\prime =xy-\text{ln}\phantom{\rule{0.1em}{0ex}}x,y\left(1\right)=1$

$\left(1+{x}^{2}\right)y\prime =y-1,y\left(0\right)=0$

$y=1-{e}^{{\text{tan}}^{-1}x}$

$xy\prime =y+2x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}x,y\left(1\right)=5$

$\left(2+x\right)y\prime =y+2+x,y\left(0\right)=0$

$y=\left(x+2\right)\text{ln}\phantom{\rule{0.1em}{0ex}}\left(\frac{x+2}{2}\right)$

$y\prime =xy+2x{e}^{x},y\left(0\right)=2$

$\sqrt{x}y\prime =y+2x,y\left(0\right)=1$

$y=2{e}^{2\sqrt{x}}-2x-2\sqrt{x}-1$

$y\prime =2y+x{e}^{x},y\left(0\right)=-1$

A falling object of mass $m$ can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant $k.$ Set up the differential equation and solve for the velocity given an initial velocity of $0.$

$v\left(t\right)=\frac{gm}{k}\left(1-{e}^{\text{−}kt\text{/}m}\right)$

Using your expression from the preceding problem, what is the terminal velocity? ( Hint: Examine the limiting behavior; does the velocity approach a value?)

[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8$ m/s 2 and the proportionality constant is $4?$

$40.451$ seconds

A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant $k.$ Set up the differential equation and solve for the velocity.

Using your expression from the preceding problem, what is the terminal velocity? ( Hint: Examine the limiting behavior: Does the velocity approach a value?)

$\sqrt{\frac{gm}{k}}$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul