Here
can be an arbitrary (positive or negative) constant. This leads to a general method for solving a first-order linear differential equation. We first multiply both sides of
[link] by the
integrating factor
This gives
Since
was previously calculated, we are now finished. An important note about the integrating constant
It may seem that we are inconsistent in the usage of the integrating constant. However, the integral involving
is necessary in order to find an integrating factor for
[link] . Only one integrating factor is needed in order to solve the equation; therefore, it is safe to assign a value for
for this integral. We chose
When calculating the integral inside the brackets in
[link] , it is necessary to keep our options open for the value of the integrating constant, because our goal is to find a general family of solutions to
[link] . This integrating factor guarantees just that.
Problem-solving strategy: solving a first-order linear differential equation
Put the equation into standard form and identify
and
Calculate the integrating factor
Multiply both sides of the differential equation by
Integrate both sides of the equation obtained in step
and divide both sides by
If there is an initial condition, determine the value of
Solving a first-order linear equation
Find a general solution for the differential equation
Assume
To put this differential equation into standard form, divide both sides by
Therefore
and
The integrating factor is
Multiplying both sides of the differential equation by
gives us
Integrate both sides of the equation.
There is no initial value, so the problem is complete.
Applications of first-order linear differential equations
We look at two different applications of first-order linear differential equations. The first involves air resistance as it relates to objects that are rising or falling; the second involves an electrical circuit. Other applications are numerous, but most are solved in a similar fashion.
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Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
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Abdullahi
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Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
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