topic
• basic concept• first order differential equation• second order linear differential equation with
constant coefficients• the Laplace transform• Fourier series
what is differential equation?
equation involving an unknown function and its derivatives
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
there are two types of differential equations (DE)
ordinary differential equation
“if the unknown function depends on only one independent variable”
partial differential equation
“if the unknown function depends on two or more independent variables”
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
order
“the order of a DE is the order of the highest derivative in the equation”
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
notation
“you can represented the order by using a simple notation”
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
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
linearity
“a crucial classification of differential equations is whether they are linear or nonlinear”
linear𝑑𝑦𝑑𝑥 + 𝑦=sin2𝑥
𝑑2 𝑦𝑑𝑥2
+3 𝑥2 𝑦=0
𝑥3 𝑑3 𝑦
𝑑 𝑥3+3 𝑥 𝑑𝑦𝑑𝑥 −5 𝑦=𝑒𝑥
non linear
( 𝑑𝑦𝑑𝑥 )2
+ 𝑦=𝑠𝑖𝑛𝑥
𝑑2 𝑦𝑑𝑥2
+3 𝑥 𝑦2=0
(1+ 𝑦)𝑑3 𝑦𝑑𝑥3
=sin 𝑦
Got the idea? No?
• We try to look again
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)
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)
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”
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.
example
• Show that
is a solution to
Solution
• We’ll need the first and second derivatives to do this
Solution
• Plug these as well as the function into the differential equation.
Solution
• So, does satisfy the differential equation and hence is a solution.
• Why then did I include the condition that x > 0?
RECALL BACK
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
Initial Value Problem
• Initial condition(s) is a condition or set of conditions that help you to solve the differential equation problem
Initial Condition(s) example
• Show that
is a solution to
where
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)
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
Actual Solution of DE
• What is the actual solution to the following IVP?
Is the general solution to
where
Actual Solution of DE
Put in
So, the actual solution to the IVP is
Implicit/Explicit Solution
Explicit solution“is any solution that is given in the form ”
Implicit solution“any solution that is not in explicit form”
Explicit function
• When an equation can be written in the form , It is said to be an explicit function of x.
• Examples
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
Homogeneous Function
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
Homogenity
• Homogenity
• HOMOGENEOUS, 2ND ORDER, AND LINEAR
• NON HOMOGENEOUS, 3RD ORDER, AND NON
LINEAR
Top Related