Differential equation hamzah asyrani sulaiman at .
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Transcript of Differential equation hamzah asyrani sulaiman at .
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topic
• basic concept• first order differential equation• second order linear differential equation with
constant coefficients• the Laplace transform• Fourier series
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what is differential equation?
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equation involving an unknown function and its derivatives
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example𝑑𝑦𝑑𝑥 =3 𝑥−1
𝑥2 𝑑2 𝑦
𝑑 𝑥2+( 𝑑𝑦𝑑𝑥 )
2
+2 𝑦= ln𝑥
𝑑3 𝑦𝑑 𝑥3
−4 𝑥 𝑑2 𝑦
𝑑 𝑥2=cos2 𝑥
𝜕 𝑦𝜕𝑡 +
𝜕 𝑦𝜕 𝑥=4 𝑡2+2𝑥
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there are two types of differential equations (DE)
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ordinary differential equation
“if the unknown function depends on only one independent variable”
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partial differential equation
“if the unknown function depends on two or more independent variables”
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𝑑𝑦𝑑𝑥 =3 𝑥−1
𝑥2 𝑑2 𝑦
𝑑𝑥2+( 𝑑𝑦𝑑𝑥 )
2
+2 𝑦= ln𝑥
𝑑3 𝑦𝑑 𝑥3
−4 𝑥 𝑑2 𝑦
𝑑𝑥2=cos2 𝑥
𝜕 𝑦𝜕𝑡 +
𝜕 𝑦𝜕 𝑥=4 𝑡2+2𝑥
ordinary differential equation
partial differential equation
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order
“the order of a DE is the order of the highest derivative in the equation”
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order?𝑑𝑦𝑑𝑥=3 𝑥−1
𝑥2 𝑑2 𝑦
𝑑 𝑥2+( 𝑑𝑦𝑑𝑥 )
2
+2 𝑦= ln𝑥
𝑑3 𝑦𝑑 𝑥3
−4 𝑥 𝑑2 𝑦
𝑑 𝑥2=cos2 𝑥
𝜕 𝑦𝜕𝑡 +
𝜕 𝑦𝜕 𝑥=4 𝑡2+2𝑥
1st order
2nd order
1st order
3rd order
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notation
“you can represented the order by using a simple notation”
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notation
𝑦 ′=𝑑𝑦𝑑𝑥=𝑦 ′ (𝑥 )
𝑦 ′ ′= 𝑑2 𝑦𝑑 𝑥2
=𝑦 ′ ′ (𝑥)
𝑦 𝑛=𝑑𝑛 𝑦𝑑𝑥𝑛=𝑦𝑛 (𝑥)
If the independent variable is
𝑦 ′=𝑑𝑦𝑑𝑡 =𝑦 ′ (𝑡)
𝑦 ′ ′= 𝑑2 𝑦𝑑𝑡 2
=𝑦 ′ ′ (𝑡)
𝑦 𝑛=𝑑𝑛 𝑦𝑑𝑡𝑛
=𝑦𝑛 (𝑡)
If the independent variable is
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Linear Differential Equations
• A linear differential equation is any differential equation that can be written in the following form
𝑎𝑛 (𝑡 ) 𝑦 (𝑛 ) (𝑡 )+𝑎𝑛−1 (𝑡 ) 𝑦 (𝑛−1 ) (𝑡 )+…+𝑎1 (𝑡 ) 𝑦 ′ (𝑡 )+𝑎0 (𝑡 ) 𝑦 (𝑡 )=𝑔 (𝑡)
*Important:-no products of the function,
y(t), and its derivatives and neither the function or its derivatives occur to any power other than the first power
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linearity
“a crucial classification of differential equations is whether they are linear or nonlinear”
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linear𝑑𝑦𝑑𝑥 + 𝑦=sin2𝑥
𝑑2 𝑦𝑑𝑥2
+3 𝑥2 𝑦=0
𝑥3 𝑑3 𝑦
𝑑 𝑥3+3 𝑥 𝑑𝑦𝑑𝑥 −5 𝑦=𝑒𝑥
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non linear
( 𝑑𝑦𝑑𝑥 )2
+ 𝑦=𝑠𝑖𝑛𝑥
𝑑2 𝑦𝑑𝑥2
+3 𝑥 𝑦2=0
(1+ 𝑦)𝑑3 𝑦𝑑𝑥3
=sin 𝑦
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Got the idea? No?
• We try to look again
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why linear?
𝑑𝑦𝑑𝑥 + 𝑦=sin2𝑥
𝑑2 𝑦𝑑𝑥2
+3 𝑥2 𝑦=0
𝑥3 𝑑3 𝑦
𝑑 𝑥3+3 𝑥 𝑑𝑦𝑑𝑥 −5 𝑦=𝑒𝑥
It has a form just like the previous one, we have:-y’(t), y(t)
It has y’’(t), and y(t)
It has y’’’(t), y’(t), and y(t)
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why non linear?
( 𝑑𝑦𝑑𝑥 )2
+ 𝑦=𝑠𝑖𝑛𝑥
𝑑2 𝑦𝑑𝑥2
+3 𝑥 𝑦2=0
(1+ 𝑦)𝑑3 𝑦𝑑𝑥3
=sin 𝑦
( 𝑦 ’ (𝑡 ) )2
( 𝑦 (𝑡 ) )2
𝑦 (𝑡 ) 𝑦 ′ ′ ′ (𝑡)
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how to detect?
“the dependent variable y and all its derivatives are of the first degree”
“the coefficient depends on constants or independent variables or both of them”
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Solution to a differential equation
• A solution to a differential equation on an interval is any function which satisfies the differential equation in question on the interval .
• It is important to note that solutions are often accompanied by intervals and these intervals can impart some important information about solution.
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example
• Show that
is a solution to
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Solution
• We’ll need the first and second derivatives to do this
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Solution
• Plug these as well as the function into the differential equation.
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Solution
• So, does satisfy the differential equation and hence is a solution.
• Why then did I include the condition that x > 0?
RECALL BACK
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Solution
• To see why recall that
• In this form, it is clear that we’ll need to avoid x=0 at the least as this would give division by zero.
• And then, x > 0 (forever).
𝑦 (𝑥 )=𝑥− 23= 1
√𝑥3
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Initial Value Problem
• Initial condition(s) is a condition or set of conditions that help you to solve the differential equation problem
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Initial Condition(s) example
• Show that
is a solution to
where
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Interval of Validity
• The interval of validity for an Initial Value Problem with initial condition(s)
And/or
Is the largest possible interval on which the solution is valid and contains
We will see it when we actually solve the differential equation (trust me I will tell you)
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General Solution of DE
• The general solution to a differential equation is the most general form that the solution can take and doesn’t take any initial conditions into account.
• Just a regular DE
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Actual Solution of DE
• What is the actual solution to the following IVP?
Is the general solution to
where
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Actual Solution of DE
Put in
So, the actual solution to the IVP is
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Implicit/Explicit Solution
Explicit solution“is any solution that is given in the form ”
Implicit solution“any solution that is not in explicit form”
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Explicit function
• When an equation can be written in the form , It is said to be an explicit function of x.
• Examples
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Implicit function
• Sometimes with equations involving, say , y and, it is impossible to make the subject of the formula. This is called implicit function
• Examples
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Homogeneous Function
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How to detect homogeneous or not
Given that Then, we put for any power of x and y
Separate it
Arrange it back so is the only remainder
Then, it can be called as homogeneous function as we still got the same equation as
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Homogenity
• Homogenity
• HOMOGENEOUS, 2ND ORDER, AND LINEAR
• NON HOMOGENEOUS, 3RD ORDER, AND NON
LINEAR