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Ordinary differential equations Ordinary differential equations Denitions
Linear differential equations and systems Linear differential equations and systems Existence and uniqueness of solutions
Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems
1. Ordinary differential equations
An ordinary differential equation of order n is an equation of
the form
dny =f x,y,dy,...,dn1y . (1)
Chapters 1-2-4: Ordinary Differential Equations dxn dx dxn1
Sections 1.1, 1.7, 2.2, 2.6, 2.7, 4.2 & 4.3 Asolution to this differential equation is an n-times
differentiable function y(x) which satises (1).
Example: Consider the differential equation
y′′ 2y′ + y =0.
What is the order of this equation?
x x
Are y (x)=e and y (x)=xe solutions of this differential
1 2
equation?
Are y (x)andy (x) linearly independent?
1 2
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
Ordinary differential equations Denitions Ordinary differential equations Denitions
Linear differential equations and systems Existence and uniqueness of solutions Linear differential equations and systems Existence and uniqueness of solutions
Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems
Initial and boundary conditions 2. Existence and uniqueness of solutions
An initial condition is the prescription of the values of y and Equation (1) may be written as a rst-order system
of its (n 1)st derivatives at a point x0, dY =F(x,Y) (3)
n1 dx
dy d y
y(x )=y , (x )=y ,... (x )=y , (2) dy d2y dn1y T
0 0 dx 0 1 dxn1 0 n1 by setting Y = y, , , ···, .
dx dx dxn1
where y0, y1, ... yn1 are given numbers. Existence and uniqueness of solutions: if F in (3) is
Boundary conditions prescribe the values of linear continuously differentiable in the rectangle
combinations of y and its derivatives for two different values R ={(x,Y), |x x | < a, ||Y Y || < b, a,b > 0},
0 0
of x. then the initial value problem
In MATH 254, you saw various methods to solve ordinary dY
=F(x,Y), Y(x )=Y ,
differential equations. Recall that initial or boundary dx 0 0
conditions should be imposed after the general solution of a has a solution in a neighborhood of (x ,Y ). Moreover, this
0 0
differential equation has been found. solution is unique.
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
Ordinary differential equations Denitions Ordinary differential equations Denitions
Linear differential equations and systems Existence and uniqueness of solutions Linear differential equations and systems Existence and uniqueness of solutions
Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems
Existence and uniqueness of solutions (continued) Existence and uniqueness for linear systems
Examples: Consider a linear system of the form
Does the initial value problem
′′ ′ ′ dY =A(x)Y +B(x),
y 2y +y =0, y(0) = 1, y (0) = 0 dx
have a solution near x =0, y =1, y′ = 0? If so, is it unique? where Y and B(x)aren×1 column vectors, and A(x)isan
Does the initial value problem n×nmatrix whose entries may depend on x.
y′ = √y, y(0) = y0 Existence and uniqueness of solutions: If the entries of the
have a unique solution for all values of y ? matrix A(x) and of the vector B(x) are continuous on some
0 open interval I containing x , then the initial value problem
Does the initial value problem 0
′ 2 dY =A(x)Y +B(x), Y(x )=Y
y =y , y(1) = 1 dx 0 0
have a solution near x =1,y = 1? Does this solution exist for has a unique solution on I.
all values of x?
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
Ordinary differential equations Denitions Ordinary differential equations General facts
Linear differential equations and systems Existence and uniqueness of solutions Linear differential equations and systems Homogeneous linear equations with constant coefficients
Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems Homogeneous linear systems with constant coefficients
Existence and uniqueness for linear systems (continued) 3. Linear differential equations and systems
The general solution of a homogeneous linear equation of
Examples: order n is a linear combination of n linearly independent
Apply the above theorem to the initial value problem solutions.
y′′ 2y′ + y =3x, y(0) = 1, y′(0) = 0 As a consequence, if we have a method to nd n linearly
independent solutions, then we know the general solution.
Does the initial value problem In MATH 254, you saw methods to nd linearly independent
y(4) x3y′′ +3y =0, solutions of homogeneous linear ordinary differential equations
y(0) = 1, y′(0) = 1, y′′(0) = 0, y(3)(0) = 0 with constant coefficients.
have a unique solution on the interval [1,1]? This includes linear equations of the form ay′′ + by′ + cy =0,
and linear systems of the form dY = AY,whereA is an
dx n
n×nconstant matrix and Y(x) is a column vector in R .
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
Ordinary differential equations General facts Ordinary differential equations General facts
Linear differential equations and systems Homogeneous linear equations with constant coefficients Linear differential equations and systems Homogeneous linear equations with constant coefficients
Nonhomogeneous linear equations and systems Homogeneous linear systems with constant coefficients Nonhomogeneous linear equations and systems Homogeneous linear systems with constant coefficients
Linear differential equations and systems (continued) Linear differential equations and systems (continued)
Aset{y1(x),y2(x),···,yn(x)} of n functions is linearly n
The Wronskian of n vectors Y (x), Y (x), ···, Y (x)inR
independent if its Wronskian is different from zero. 1 2 n
is given by
Similarly, a set of n vectors {Y (x),Y (x),···,Y (x)} in Rn
1 2 n
W(Y ,Y ,···,Y )=det([Y Y ··· Y ]),
is linearly independent if its Wronskian is different from zero. 1 2 n 1 2 n
The Wronskian of n functions y (x), y (x), ···, y (x)is where [Y Y ··· Y ] denotes the n ×n matrix whose
1 2 n 1 2 n
given by columns are Y (x), Y (x), ···, Y (x).
1 2 n
y1 y2 ··· yn Finding n linearly independent solutions to a homogeneous
′ ′ ′
y1 y2 ··· yn linear differential equation or system of order n, is equivalent
′′ ′′ ′′
y1 y2 ··· yn to nding a basis for the set of solutions.
W(y ,y ,···,y )=
1 2 n
. . . .
. . .. .
. . . The next two slides summarize how to nd linearly
y1(n1) y2(n1) ··· yn(n1) independent solutions in two particular cases.
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
Ordinary differential equations General facts Ordinary differential equations General facts
Linear differential equations and systems Homogeneous linear equations with constant coefficients Linear differential equations and systems Homogeneous linear equations with constant coefficients
Nonhomogeneous linear equations and systems Homogeneous linear systems with constant coefficients Nonhomogeneous linear equations and systems Homogeneous linear systems with constant coefficients
Homogeneous linear equations with constant coefficients Homogeneous linear systems with constant coefficients
To nd the general solution to an ordinary differential equation of dY
the form ay′′ +by′ +cy =0,wherea,b,c ∈ R, proceed as follows. To nd the general solution of the linear system dx = AY,where
Ais an n×n matrix with constant coefficients, proceed as follows.
1 Find the characteristic equation, aλ2 + bλ + c = 0 and solve 1 Find the eigenvalues and eigenvectors of A.
for the roots λ1 and λ2. 2 If the matrix has n linearly independent eigenvectors
2 U ,U ,···,U , associated with the eigenvalues
2 If b 4ac > 0, then the two roots are real and the general 1 2 n
λ x λ x λ ,λ,···,λ , then the general solution is
solution is y = C e 1 +C e 2 . 1 2 n
1 2 λ x λ x λ x
Y =C U e 1 +C U e 2 +···+C U e n ,
1 1 2 2 n n
3 If b2 4ac < 0 the two roots are complex conjugate of one where the eigenvalues λ may not be distinct from one
i
another and the general solution is of the form another, and the C s, λ s and U s may be complex.
i i i
y = eαx (C cos(βx)+C sin(βx)),whereα = ℜe(λ )=b,
1 √ 2 1 2a If A has real coefficients, then the eigenvalues of A are either real
and β = ℑm(λ )= 4acb2. or come in complex conjugate pairs. If λ = λ , then the
1 2a i j
corresponding eigenvectors U and U are also complex conjugate
4 If b2 4ac =0, then there is a double root λ = b , and the i j
λx 2a of one another.
general solution is y =(C1 + C2x)e .
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
Ordinary differential equations Ordinary differential equations
Linear differential equations and systems Linear differential equations and systems
Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems
4. Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems (continued)
The general solution y to a non-homogeneous linear equation
of order n is of the form
y(x)=yh(x)+yp(x),
where yh(x) is the general solution to the corresponding In MATH 254, you saw methods to nd particular solutions to
homogeneous equation and yp(x) is a particular solution to non-homogeneous linear equations and systems of equations.
the non-homogeneous equation.
Similarly, the general solution Y to a linear system of You should review these methods and make sure you know
equations dY = A(x)Y +B(x)isoftheform how to apply them.
dx
Y(x)=Y (x)+Y (x),
h p
where Y (x) is the general solution to the homogeneous
h
system dY = A(x)Y and Y (x) is a particular solution to the
dx p
non-homogeneous system.
Chapters 1-2-4: Ordinary Differential Equations Chapters 1-2-4: Ordinary Differential Equations
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