These equations are nonlinear because of terms like
${\left({y}^{\prime}\right)}^{4},{y}^{3},$ etc. Due to these terms, it is impossible to put these equations into the same form as
[link] .
Our main goal in this section is to derive a solution method for equations of this form. It is useful to have the coefficient of
${y}^{\prime}$ be equal to
$1.$ To make this happen, we divide both sides by
$3{x}^{2}-4.$
This is called the
standard form of the differential equation. We will use it later when finding the solution to a general first-order linear differential equation. Returning to
[link] , we can divide both sides of the equation by
$a\left(x\right).$ This leads to the equation
Now define
$p\left(x\right)=\frac{b\left(x\right)}{a\left(x\right)}$ and
$q\left(x\right)=\frac{c\left(x\right)}{a\left(x\right)}.$ Then
[link] becomes
${y}^{\prime}+p(x)y=q(x).$
We can write any first-order linear differential equation in this form, and this is referred to as the standard form for a first-order linear differential equation.
Writing first-order linear equations in standard form
Put each of the following first-order linear differential equations into standard form. Identify
$p\left(x\right)$ and
$q\left(x\right)$ for each equation.
$y\prime =3x-4y$
$\frac{3xy\prime}{4y-3}=2$ (here
$x>0)$
$y=3y\prime -4{x}^{2}+5$
Add
$4y$ to both sides:
$y\prime +4y=3x.$
In this equation,
$p\left(x\right)=4$ and
$q\left(x\right)=3x.$
Multiply both sides by
$4y-3,$ then subtract
$8y$ from each side:
Finally, divide both sides by
$3x$ to make the coefficient of
$y\prime $ equal to
$1\text{:}$
$y\prime -\frac{8}{3x}y=-\frac{2}{3x}.$
This is allowable because in the original statement of this problem we assumed that
$x>0.$ (If
$x=0$ then the original equation becomes
$0=2,$ which is clearly a false statement.)
In this equation,
$p\left(x\right)=-\frac{8}{3x}$ and
$q\left(x\right)=-\frac{2}{3x}.$
Subtract
$y$ from each side and add
$4{x}^{2}-5\text{:}$
We now develop a solution technique for any first-order linear differential equation. We start with the standard form of a first-order linear differential equation:
$y\prime +p\left(x\right)y=q\left(x\right).$
The first term on the left-hand side of
[link] is the derivative of the unknown function, and the second term is the product of a known function with the unknown function. This is somewhat reminiscent of the power rule from the
Differentiation Rules section. If we multiply
[link] by a yet-to-be-determined function
$\mu \left(x\right),$ then the equation becomes
Matching term by term gives
$y=f\left(x\right),g\left(x\right)=\mu \left(x\right),$ and
${g}^{\prime}\left(x\right)=\mu \left(x\right)p\left(x\right).$ Taking the derivative of
$g\left(x\right)=\mu \left(x\right)$ and setting it equal to the right-hand side of
${g}^{\prime}\left(x\right)=\mu \left(x\right)p\left(x\right)$ leads to
This is a first-order, separable differential equation for
$\mu \left(x\right).$ We know
$p(x)$ because it appears in the differential equation we are solving. Separating variables and integrating yields
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
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?
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
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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?