1st order differential equations
TRANSCRIPT
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First Order Ordinary Linear Differential Equations
• Ordinary Differential equations does not include partial derivatives.
• A linear first order equation is an equation that can be expressed in the form
Where p and q are functions of x
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Types Of Linear DE:
1. Separable Variable2. Homogeneous Equation3. Exact Equation4. Linear Equation
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The first-order differential equation
,dy f x ydx
is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
(1)
Separable Variable
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Suppose we can write the above equation as
)()( yhxgdxdy
We then say we have “ separated ” the variables. By taking h(y) to the LHS, the equation becomes
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1 ( )( )
dy g x dxh y
Integrating, we get the solution as
1 ( )( )
dy g x dx ch y
where c is an arbitrary constant.
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Example 1. Consider the DE ydxdy
Separating the variables, we get
dxdyy
1
Integrating we get the solution askxy ||ln
or ccey x , an arbitrary constant.
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Homogeneous equationsDefinition A function f(x, y) is said to be homogeneous of degree n in x, y if
),(),( yxfttytxf n for all t, x, yExamples 22 2),( yxyxyxf is homogeneous of degree
)sin(),(x
xyxyyxf
is homogeneous of degree
2.
0.
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A first order DE 0),(),( dyyxNdxyxM
is called homogeneous if ),(,),( yxNyxMare homogeneous functions of x and y of the same degree.
This DE can be written in the form
),( yxfdxdy
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The substitution y = vx converts the given equation into “variables separable” form and hence can be solved. (Note that v is also a new variable)
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Working Rule to solve a HDE:1. Put the given equation in the form
.Let .3 zxy
)1(0),(),( dyyxNdxyxM
2. Check M and N are Homogeneous function of the same degree.
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5. Put this value of dy/dx into (1) and solve the equation for z by separating the variables.
6. Replace z by y/x and simplify.
dxdzxz
dxdy
4. Differentiate y = z x to get
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EXACT DIFFERENTIAL EQUATIONS
A first order DE 0),(),( dyyxNdxyxMis called an exact DE if
M Ny x
The solution id given by:
cNdyMdx(Terms free from ‘x’)
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Solution is:
cyxy
cdyy
ydx
ln2
2
cNdyMdx
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Integrating FactorsDefinition: If on multiplying by (x, y), the DE
0M dx N dy
becomes an exact DE, we say that (x, y) is an Integrating Factor of the above DE
2 2
1 1 1, ,xy x y
are all integrating factors of
the non-exact DE 0y dx x dy
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( )
M Ny x g x
N
( )g x dxe
is an integrating factor of the given DE
0M dx N dy
Rule 1: I.F is a function of ‘x’ alone.
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Rule 2: I.F is a function of ‘y’ alone.
If , a function of y alone,
then( )h y dy
e is an integrating factor of the given DE.
xN
yM
M yh
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is an integrating factor of the given DE
0M dx N dy
Rule 3: Given DE is homogeneous.
NyMx1
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Rule 4: Equation is of the form of
021 xdyxyfydxxyf
Then,
NyMx
1
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Linear EquationsA linear first order equation is an equation that can be expressed in the form
1 0( ) ( ) ( ), (1)dya x a x y b xdx
where a1(x), a0(x), and b(x) depend only on the independent variable x, not on y.
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We assume that the function a1(x), a0(x), and b(x) are continuous on an interval and that a1(x) 0on that interval. Then, on dividing by a1(x), we can rewrite equation (1) in the standard form
( ) ( ) (2)dy P x y Q xdx
where P(x), Q(x) are continuous functions on the interval.
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Rules to solve a Linear DE:1. Write the equation in the standard form
( ) ( )dy P x y Q xdx
2. Calculate the IF (x) by the formula
( ) exp ( )x P x dx 3. Multiply the equation by (x).
4. Integrate the last equation.
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Applications
)mTα(TdtdT
αdt)T(T
dT
m
•Cooling/Warming lawWe have seen in Section 1.4 that the mathematical formulation of
Newton’s empirical law of cooling of an object in given by the linear first-order differential equation
This is a separable differential equation. We have
or ln|T-Tm |=t+c1
or T(t) = Tm+c2et
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In Series Circuits
i(t), is the solution of the differential equation.
)(tERidtdi
dtdqi
)t(EqC1
dtdqR
Since
It can be written as