Differential Equations - ODE of First Order
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Differential Equations
Ordinary Differential Equations of Order OneOrder One
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Ordinary Differential Equations of Order One
1. Variable-Separable Equations
2. HomogeneousEquations2. HomogeneousEquations
3. Exact Equations
4. Linear Differential Equations of the FirstOrder
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General Form of Ordinary Differential Equations of the First Order
Consider the form
where bothM andN can be functions ofx, y, orbothx andy.
( ) ( ), , 0M x y dx N x y dy+ =
bothx andy.
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Variable-Separable Equations
Given
If this equation can be expressed as
( ) ( ) 0A x dx B y dy+ =
( ) ( ), , 0M x y dx N x y dy+ =
then it is a variable-separable equation.
( ) ( ) 0A x dx B y dy+ =
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Examples
Problems:
( )2
1. sin sin cos cos 0
2. cos sin
3. cos tan 0
x ydx x ydy
dr b dr r d
x ydx ydy
θ θ θ+ =
= +
+ =
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Examples
Answers:
( )2 2 2
1. sin cos
2. 1 cos
3. tan
y C x
r C b
x y C
θ=
= −
+ =
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Homogeneous Equations
Given
If each termof the equation has a total degree ofn (sum of exponents of all the variables in aterm), then the equation is a homogeneous
( ) ( ), , 0M x y dx N x y dy+ =
term), then the equation is a homogeneousdifferential equation of degreen.
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Homogeneous Equations
To solve a homogeneous equation, one maychooseto substitutechooseto substitute
or
An advantagemay be gained if M has fewer
x vy dx vdy ydv= = +
y vx dy vdx xdv= = +An advantagemay be gained if M has fewer
terms thanN andx = vy is chosen. Same goesfor N has fewer terms andy = vx. The resultingequation becomes variable-separable.
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Homogeneous Equations
Theorem 1. If M(x,y) and N(x,y) are bothhomogeneousand of the same degree, thehomogeneousand of the same degree, thefunction M(x,y)/N(x,y) is homogeneous ofdegree zero.
Theorem 2. Iff(x,y) is homogeneous of degreezeroin x andy, f(x,y) is aunctionof y/x alone.zeroin x andy, f(x,y) is aunctionof y/x alone.
( ) ( ) ( ) ( )0, , 1, 1,f x y f x vx x f v f v= = =
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Examples
Problems:
( )2 21. 3 3 2 0x y dx xydy+ − =( )( )
( )
2 2
2 2
1. 3 3 2 0
2. 3 0
3. csc 0yx
x y dx xydy
xydx x y dy
x y dx xdy
+ − =
+ + =
− + = ( )3. csc 0xx y dx xdy − + =
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Examples
Answers:
( )3 2 21. 9x C x y= +( )( )
( )
3 2 2
32 2 2 2
1. 9
2. 4
3. ln cos yxc x
x C x y
y x y C
= +
+ =
= ( )3. ln cosc x=
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Exact Equations
Given
If the following partial differentials are equal,
( ) ( ), , 0M x y dx N x y dy+ =
M N∂ ∂=∂ ∂
then it is an exact differential equation.y kx k
y x ==
=∂ ∂
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Exact Equations
To solve an exact differential equation, set
Then solve for F by integrating one of thefunctions with respect to its partial differentialindependentvariable (with the other variable
or F F
M Nx y
∂ ∂= =∂ ∂
independentvariable (with the other variabletreated as constant.
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Exact Equations
If M was initially chosen, setT’(y) with functionterms of N with y variablesonly. If N wasterms of N with y variablesonly. If N wasinitially chosen, setT’(x) with function termsof M with x variables only. SolveT byintegrating the function obtained.
Thesolutionis thenThesolutionis then( ) ( )
( ) ( )
,
or
,
F x y T x C
F x y T y C
+ =
+ =
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Exact Equations
Tips and tricks: the shortcuts
or
( ) ( ),y k
M x y dx N y dy C=
+ =∫ ∫
( ) ( ),x k
M x dx N x y dy C=
+ =∫ ∫
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Examples
Problems:
( ) ( )( ) ( )( ) ( )
3 2 3 2
2 2
1. 2 cos cos 0
2. 0
3. 2 tan sec 0
x y xy dx x xy dy
w wz z dw z w z w dz
xy y dx x x y dy
+ + =
+ − + + − =
− + − =
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Examples
Answers:
( )( )
2
22 2
2
1. sin
2. 4
3. tan
x xy C
w z wz C
x y x y C
+ =
+ = +
− =
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Linear Differential Equation of the First Order
Given
( ) ( )+ =If this equation can be expressed as
( ) ( ), , 0M x y dx N x y dy+ =
( ) ( )
( ) ( ) or
dy yP x dx Q x dx+ =
+ =then it is a linear differential equation of the first
order.
( ) ( )dx xP y dy Q y dy+ =
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Linear Differential Equation of the First Order
To solve the linear differential equation of thefirst order,determinetheintegratingfactorbyfirst order,determinetheintegratingfactorby
Then solve the equation
( ) ( ) or
P x dx P y dyv e v e∫ ∫= =
( )vy vQ x dx C= +∫
( ) or
vx vQ y dy C= +∫
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Examples
Problems:
( )( )
2
1. ' csc cot
2. 2
3. 1 2 tan 0
y x y x
y y x dy dx
dx x y dy
= −
− =
− + =
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Examples
Answers:
22
2
1. sin
2. 1
3. 2 cos sin cos
y
y x x C
x y Ce
x y y y y C
−
= +
= − += + +