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Solution of
BernoullisEquationsAcademic Resource Center
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Contents
First order ordinary differential equation
Linearity of Differential Equations
Typical Form of Bernoullis Equation
Examples of Bernoullis Equations
Method of Solution
Bernoulli Substitution
Example Problem
Practice Problems
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First Order differential equations
A differential equation having a first derivative as the highest
derivative is a first order differential equation.
If the derivative is a simple derivative, as opposed to a partial
derivative, then the equation is referred to as ordinary.
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Linearity of Differential Equations
The terminology linear derives from the description of a line.
A line, in its most general form, is written as + = Similarly, if a differential equation is written as +
= () Then this equation is termed linear, as the highest power of
and y is 1. They are analogous to x and y in the equation of a
line, hence the term linear.
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Typical form of Bernoullis
equation The Bernoulli equation is a Non-Linear differential equation of
the form + = ()
Here, we can see that since y is raised to some power n where
n1.
This equation cannot be solved by any other method like
homogeneity, separation of variables or linearity.
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Examples
Here are a few examples of Bernoullis Equation
+ =
= + 1
+ =
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Method of Solution
The first step to solving the given DE is to reduce it to the
standard form of the Bernoullis DE. So, divide out the whole
expression to get the coefficient of the derivative to be 1.
Then we make a substitution = This substitution is central to this method as it reduces a non-
linear equation to a linear equation.
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Bernoulli Substitution
So if we have = ,then
= (1 )
From this, replace all the ys in the equation in terms of u and
replace in terms ofand u. This will reduce the whole equation to a linear differential
equation.
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Example Problem
Let us try to solve the given differential equation
= 2First of all, we rearrange it so that the derivative and first power
of y are on one side.
2 = Then we divide out by to make the co-efficient of thederivative equal to 1.
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Contd.
Then, that leaves us with
2 =
1
Here, we identify the power on the variable y to be 2. Therefore,
we make the substitution = = Thus,
=
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Contd.
Then, knowing that:
=
and =
We have =
and
=
We substitute these expressions into our Bernoullis equation and
that gives us
1
2
1 =
1
1
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Contd.
Once we have reached this stage, we can multiply out by and then the DE simplifies to
+
2 =
1
This is now a linear differential equation in u and can be solvedby the use of the integrating factor. Thats the rational behind the
use of the specialized substitution.
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Contd.
Proceeding, we identify = , we have the integratingfactor
= = ln() = Then multiplying through by , we get + 2 = 1
Which simplifies to
= 1
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Contd.
We now integrate both sides,
=1
This gives = + = 1 +
We have now obtained the solution to the simplified DE. All we
have to do now is to replace =
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Contd.
Our final answer will therefore look like
= 1 +
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Practice Problems
Try practicing these problems to get the hang of the method.
You can also try solving the DEs given in the examples section.
1. 1 + = 2.
= (
1)
3. 3 1 + = 2( 1)
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References
A first Course in Differential Equations 9th Ed., Dennis Zill.
Elementary Differential Equations, Martin & Reissner
Differential and Integral Calculus Vol 2, N. Piskunov
Workshop developed by Abhiroop Chattopadhyay