Use separation of variables to solve a differential equation.
Solve applications using separation of variables.
We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section.
Separation of variables
We start with a definition and some examples.
Definition
A
separable differential equation is any equation that can be written in the form
$y\prime =f\left(x\right)g\left(y\right).$
The term ‘separable’ refers to the fact that the right-hand side of the equation can be separated into a function of
$x$ times a function of
$y.$ Examples of separable differential equations include
The second equation is separable with
$f\left(x\right)=6{x}^{2}+4x$ and
$g\left(y\right)=1,$ the third equation is separable with
$f\left(x\right)=1$ and
$g\left(y\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}y+\text{tan}\phantom{\rule{0.1em}{0ex}}y,$ and the right-hand side of the fourth equation can be factored as
$\left(x+3\right)\left(y-2\right),$ so it is separable as well. The third equation is also called an
autonomous differential equation because the right-hand side of the equation is a function of
$y$ alone. If a differential equation is separable, then it is possible to solve the equation using the method of
separation of variables .
Problem-solving strategy: separation of variables
Check for any values of
$y$ that make
$g(y)=0.$ These correspond to constant solutions.
Rewrite the differential equation in the form
$\frac{dy}{g(y)}=f(x)dx.$
Integrate both sides of the equation.
Solve the resulting equation for
$y$ if possible.
If an initial condition exists, substitute the appropriate values for
$x$ and
$y$ into the equation and solve for the constant.
Note that Step 4. states “Solve the resulting equation for
$y$ if possible.” It is not always possible to obtain
$y$ as an explicit function of
$x.$ Quite often we have to be satisfied with finding
$y$ as an implicit function of
$x.$
Using separation of variables
Find a general solution to the differential equation
$y\prime =\left({x}^{2}-4\right)\left(3y+2\right)$ using the method of separation of variables.
Follow the five-step method of separation of variables.
In this example,
$f\left(x\right)={x}^{2}-4$ and
$g\left(y\right)=3y+2.$ Setting
$g(y)=0$ gives
$y=-\frac{2}{3}$ as a constant solution.
To solve this equation for
$y,$ first multiply both sides of the equation by
$3.$
$\text{ln}\left|3y+2\right|={x}^{3}-12x+3C$
Now we use some logic in dealing with the constant
$C.$ Since
$C$ represents an arbitrary constant,
$3C$ also represents an arbitrary constant. If we call the second arbitrary constant
${C}_{1},$ the equation becomes
$\text{ln}\left|3y+2\right|={x}^{3}-12x+{C}_{1}.$
Now exponentiate both sides of the equation (i.e., make each side of the equation the exponent for the base
$e).$
Again define a new constant
${C}_{2}={e}^{{c}_{1}}$ (note that
${C}_{2}>0)\text{:}$
$\left|3y+2\right|={C}_{2}{e}^{{x}^{3}-12x}.$
This corresponds to two separate equations:
$3y+2={C}_{2}{e}^{{x}^{3}-12x}$ and
$3y+2=\text{\u2212}{C}_{2}{e}^{{x}^{3}-12x}.$ The solution to either equation can be written in the form
$y=\frac{\mathrm{-2}\pm {C}_{2}{e}^{{x}^{3}-12x}}{3}.$ Since
${C}_{2}>0,$ it does not matter whether we use plus or minus, so the constant can actually have either sign. Furthermore, the subscript on the constant
$C$ is entirely arbitrary, and can be dropped. Therefore the solution can be written as
$y=\frac{\mathrm{-2}+C{e}^{{x}^{3}-12x}}{3}.$
No initial condition is imposed, so we are finished.
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?