MTH2003 DIFFFERENTIAL EQUATIONS Flashcards
DEFINITION: independent variable
in the function y(x), x is the independent variable
DEFINITION: dependent variable
in the function y(x), y is the dependent variable
DEFINITION: ordinary differential equation (ODE)
an equation that relates values of a function to its derivatives with
respect to one independent variable, where F(. . .) is a given function, and y(x) is the function to be
found, i.e. unknown function
DEFINITION: solution of an ODE
a function y = φ(x) is a solution of an ODE in an
interval I : {x : a < x < b} NOT= ∅, if its substitution into the ODE
produces an identity.
DEFINITION: partial differential equation (PDE)
an equation that relates values of a function to partial derivatives with
respect to more than one independent variable
DEFINITION: order of a differential equation
the order of
the highest-order derivative of the unknown function that appears
in it
DEFINITION: ODE in resolved form (standard form)
an equality of the form d^ny/dx^n = f(x, y, dy/dx, . . . , d^(n−1)y/dx^(n−1)) where f(x, y, . . .) is given, and y(x) is to be found
DEFINITION: right-hand side
the function f(x, y, . . )
DEFINITION: initial value problem (IVP) (also known as Cauchy problem or one-point boundary value problem with boundary conditions)
an ODE plus initial conditions (extra requirements on y(x))
DEFINITION: solution of the initial value problem
a function y = φ(x) is a solution of the initial
value problem
d^ny/dx^n = f (x, y, . . .), y(x0) = K0, . . .
d^(n−1)y/dx^(n−1)(x0) = Kn−1 in an interval
I : {x : a < x < b} NOT= ∅, such that x0 ∈ I,
if it is a solution of the ODE in that interval, and satisfies the
initial condition
DEFINITION: separable first-order ODE (ODE with separable variables)
an ODE right-hand side is of the special form dy/dx = f(x, y) = F(x) G(y(x))
DEFINITION: special solution
if G(y∗) = 0, then y(x) = y∗ is a special solution
ALGORITHM: test for separability
If f(x, y) = F(x)G(y) for all x, y, then f(a, b)f(c, d) ≡ f(a, d)f(c, b) for all a, b, c, d. if at least one combination of a, b, c, d violates the equality, this is a definite proof that f(x, y) is not separable (it is not possible to present it in the form F(x)G(y))
THEOREM: Peano’s existence theorem
consider the initial value problem
dy/dx = f(x, y),
y(x0) = K0,
and assume that there exists a rectangle R = (α, β) × (γ, δ) NOT= ∅,
such that
1◦ The initial condition is within this rectangle, (x0, K0) ∈ R.
2◦ The right-hand side f(x, y) is continuous for all (x, y) ∈ R.
-> Then there exists an interval of x values, I = (a, b), where x0 ∈ I,
such that initial value problem has at least one solution in
the interval I.
THEOREM: Picard’s existence and uniqueness theorem
Consider the initial value problem
dy/dx = f(x, y),
y(x0) = K0,
and assume that there exists a rectangle R = (α, β) × (γ, δ) NOT= ∅,
such that
1◦ The initial condition is within this rectangle, (x0, K0) ∈ R;
2◦ The right-hand side f(x, y) is continuous for all (x, y) ∈ R;
3◦ The partial derivative fy(x, y) is defined and continuous for
all (x, y) ∈ R.
_> Then there exists an interval of x values, I = (a, b), where x0 ∈ I,
such that initial value problem has a UNIQUE solution in the
interval I.
DEFINITION: integrating factor
h(x) = exp (integral(p(x))dx)
DEFINITION: derivative of a vector-function x(t)
dx/dt := lim[h→0] x(t + h) − x(t) / h
THEOREM: Picard’s existence and uniqueness theorem for systems
Consider the initial value problem consisting of the
ODE system and the initial conditions, and assume that
there exist such a box B in the R^N+1 = {(t, x1, . . . xN)} space,
defined by conditions B : α < t < β, α1 < x1 < β1, . . . , αN < xN <
βN, B NOT= ∅, such that
1◦ The point representing the initial conditions is inside the box,
(t0, K1 . . . KN) ∈ B,
2◦ The N right-hand sides fj(t, x1, . . . xN), j = 1, . . . , N are defined
and continuous for all (t, x) ∈ B,
3◦ The N × N partial derivatives ∂ fj(t, x1, . . . xN)/∂xk, j = 1 . . . N,
k = 1 . . . N are defined and continuous for all (t, x) ∈ B.
-> Then there exists an interval of t values, I = (a, b), containing
the initial point t0, that is t0 ∈ I, such that initial value problem has a unique solution in the interval I
DEFINITION: linear ODE system
a linear ODE system if all the righthand sides depend on all the unknowns x_k
linearly (the coefficients of the linear functions may depend on t)
THEOREM: Existence and uniqueness theorem for linear systems
If coefficients a_jk(t) and g_k(t), j = 1 . . . N, k = 1 . . . N
are continuous in an interval t ∈ (α, β), then for initial conditions such that t0 ∈ I, there exists an interval of t values, I = (a, b), containing the initial point t0, that is t0 ∈ I, such
that initial value problem has a unique solution in the interval I
DEFINTION: eigenvector and eigenvalue of matrix A
If A ∈ C^N×N is a N × N matrix and the column vector v ∈ CN satisfies
Av = λv, v NOT= 0, where λ ∈ C is a number,
then v is called an eigenvector of matrix A and λ is the eigenvalue
LEMMA: 2 If v is an eigenvector of A corresponding to eigenvalue λ,
then kv for any nonzero scalar k is also an eigenvector of A corresponding to the same λ.
Proof:
1◦ If v NOT= 0 and k NOT= 0, then kv NOT= 0
2◦ If Av = λv, then A(kv) = k(Av) = kλv = λ(kv)
That is, kv satisfies both requirements for an eigenvector.
DEFINITION: characteristic equation
det(A - λI) = 0
DEFINITION:
linear independent system of vectors
A system of m vectors x^(1), . . . x^(m) of the same dimensionality N is called linearly independent (LI), if the equality c1x^(1) + · · · + cmx^(m) = 0
is possible only if all c1 = · · · = cm = 0. If a system of vectors is
not linearly independent, it is called linearly dependent (LD).
THEOREM: linear independence of eigenvectors
Let v^1, . . . , v^m be eigenvectors, corresponding to eigenvalues λ1, . . . , λm, of matrix A ∈ C^N×N where 1 < m ≤ N and let all λj , j = 1, . . . , λm be different.
Then vectors v^1, . . . , v^m are linearly independent.
DEFINITION: generalised eigenvector (GEV) of m-th order
Let v be an eigenvector of A corresponding to eigenvalue.
Then a generalised eigenvector (GEV) of m-th order, m = 1, 2 . . . , is a vector v^(m)
satisfying:
Av^(m) = λv^(m) + v^(m−1), where v^(0) := v
DEFINTION: Jordan chain of GEV
The set of vectors (v^(0), . . . , v^(m)), is called a Jordan chain
DEFINITION: lead vector or generator of a Jordan chain in a GEV
The GEV v^(m) is called the lead vector or the generator of the chain.
DEFINITION: Jordan block or a Jordan cell
An m × m matrix of the form below is called a Jordan block or a Jordan cell of size m and eigenvalue λ. (a square matrix with zeros everywhere and then λ on the diagonal with 1’s above them on the diagonal above)
DEFINITION: Jordan matrix or a matrix in Jordan form
A Jordan matrix or a matrix in Jordan (normal/canonical)
form is a square matrix which has the following structure:
along the main diagonal, there are Jordan blocks, so that the
diagonals of the blocks coincide with the diagonal of the whole
Jordan matrix,
• the sum of sizes of the Jordan blocks makes the size of the
whole Jordan matrix,
• outside the diagonal blocks, all entries are zero.
DEFINITION: Cramer’s rule
To solve a system of simultaneous linear algebraic
equations,
a_11x_1 + a_12x_2 = b_1,
a_21x_1 + a_22x_2 = b_2,
multiply first equation by a_22, second by a_12 and take the difference;
this leads to x_1 = b1_a_22 − b_2a_12 / a_11a_22 − a_12a_21,
and x_2 = b1_a_21 − b_2a_11 / a_12a_21 − a_11a_22. Using determinants, the results are conveniently
written as
x_1 = ∆_1 / ∆ , x_2 = ∆2_ / ∆,
where
∆ = a11 a12
a21 a22,
∆1 = b1 a12
b2 a22,
∆2 = a11 b1
a21 b2DEFINITION: a homogeneous system of linear system
A homogeneous system of linear ODEs is a linear
system in which the free terms are zero, i.e. in vector notation,
dx / dt = A(t)x.
THEOREM: If x^(1)(t), . . . , x^(m)(t) are solutions, then a linear combination of those with constant coefficients, i.e.
x(t) = c1x ^(1)(t) + · · · + cmx^(m)(t)
is also a solution.
PROOF: Using for brevity Σ-notation, we write x(t) = m∑j=1 (cjx^(j)(t)), and substituting this into dx / dt = A(t)x we get, using the properties of differentiation and matrix multiplication, LHS = d/dt m∑j=1 (cjx^(j)(t)) = m∑j=1 cj d/dt x^(j)(t) = m∑j=1 cjA(t)x^(j)(t) = A(t) m∑j=1 cjx^(j)(t) = A(t)x(t) = RHS
COROLLARY: In particular, x(t) ≡ 0 is always a solution to dx / dt = A(t)x
This is the trivial solution
DEFINITION: linear independent set
Consider a set of m vector-functions f^(1)(t), . . . ,f^(m)(t),
depending on a scalar argument t and with values in R^N, all defined on an interval I = (α, β).
This set is said to be linearly independent (LI) on I, if the identity
c_1f^(1)(t) + · · · + c_mf^(m)(t) = 0 ∀t ∈ I
is possible with scalar constants c_1, . . . , c_m only if all c_1 = · · · =
c_m = 0.
Conversely, if the linear combination of functions can be identically zero on I with at least one coefficient nonzero, then
the set is said to be linearly dependent (LD) on I
THEOREM: Let x^(1)(t), . . . , x^(m)(t) be solutions of dx / dt = A(t)x on I, and
t0 ∈ I. Then these solutions are linearly dependent as vector functions on I if and only if x^(1)(t0), . . . , x^(m)(t0) are linearly dependent as vectors.
PROOF: 1
◦ One way the assertion is straightforward: LD as functions implies LD as vectors. That is, if
there is a nontrivial set of cj such that m∑j=1 (c_jx)(j)(t) = 0 for all t ∈ I, it is true for t = t0 ∈ I.
2◦ Now we prove that LD as vectors implies LD as functions. Suppose we have found a nontrivial
set of cj such that m∑j=1
(c_jx)(j)(t0) = 0.
Consider now x(t) := m∑j=1
(c_jx)(j)(t).
This is a linear combination of solutions. We have x(t0) = 0 by assumption. The trivial solution x(t) ≡ 0 satisfies this initial condition. the solution of the initial value problem is unique, we
have 0 = x(t) = m∑j=1 c_jx^(j)(t) for all t ∈ I, that is the solutions are LD as functions on I.
DEFINITION: fundamental system (FS)
A set of N linearly independent solutions x^(1)(t), . . . , x^(N)(t) of dx / dt = A(t)x is called fundamental system (FS) or fundamental set (FS) of solution
THEOREM: Fundamental system of solutions always exists
Take any t0 ∈ I and set
x^(1)(t0) = [ 1, 0, . . . , 0 ]^T, x^(2)(t0) = [ 0, 1, . . . , 0 ]^T,
. . x^(N)(t0) = [ 0, 0, . . . , 1 ]^T
.Solutions exist on I, and are LI on I since they are LI at t0 ∈ I.
DEFINITION: a fundamental matrix
A square matrix-function Ψ(t) whose columns make
a fundamental set of solutions of system dx / dt = A(t)x, is called a fundamental matrix (FM) of that system
THEOREM: A fundamental matrix Ψ(t) of dx / dt = A(t)x satisfies a matrix ODE
dΨ / dt = AΨ.
PROOF:
LHS: the j-th column of dΨ/dt is dx^(j)/dt.
RHS: the j-th column of AΨ is Ax^(j). Since x^(j) is a solution, we have dx^(j)/dt = Ax(j), that is,
that each column of the LHS equals corresponding column of the RHS.
THEOREM: Let x^(1)(t), . . . , x^(N)(t) be a fundamental set of solutions of dx / dt = A(t)x. Then for any solution x(t) of dx / dt = A(t)x, there exists a set of constants c1, . . . , cN, such that;
x(t) = N∑j=1 cjx^(j)(t),
and such set of constants is unique
PROOF: For any given solution x(t), consider a point t = t0 ∈ I.
Vectors x^(1)(t0), . . . , x^(N)(t0) make a basis in R^N. Hence, the vector x(t0) is uniquely expanded in this basis. Let c1, . . . , cN be coefficients of that expansion.
The function ∑ cjx^(j)(t) is a solution; it coincides with the given x(t) at t = t0.
Such solution is unique, hence we have x(t) = ∑ cjx^(j)(t), and the choice of cj is unique, as claimed.
THEOREM: Consider a non-homogeneous system of first-order ODEs of order N,
dx / dt = A(t)x + g(t),
and the corresponding homogeneous system
dx / dt = A(t)x.
Suppose x∗(t) is a solution and x^(1)(t), . . . , x^(N)(t) is a
fundamental set of solutions. Then x(t) is a solution if and only if it can be presented in the form:
x(t) = x∗(t) + C1x^(1)(t) + · · · + CNx^(N)(t) for some constants C1, . . . , CN.
PROOF: By assumption, both x(t) and x∗(t) are solutions; that is x’ = A(t)x + g(t) and x’∗ = A(t)x∗ + g(t).
Consider the difference y(t) = x(t) − x∗(t). We have
y’ = x’ − x’∗ = A(t)x − A(t)x∗ = A(t)(x − x∗) = A(t)y
hence y(t) is a solution.
by Theorem, y = N∑k=1 Ckx^(k)(t) (“if and only if”), and then by writing x = x∗ + y, we obtain the desired form.
GS = PI + CF
DEFINITION: particular integral
x∗(t)
DEFINITION: general solution
x(t)
DEFINITION: complementary function
∑k Ckx^(k)(t)
DEFINITION: (second order differential) linear operator
A second order differential linear operator is a mapping
y(x) |→ L[y(x)] =
a_2(x)y’‘(x) + a_1(x)y’(x) + a_0(x)y(x).
a_0,1,2(x) are fixed (“known”) functions, called coefficients of the operator, and a_2(x) NOT≡ 0.
LEMMA: For any functions y_1(x), y_2(x), . . . y_k(x) for which the operator L is defined, and any constants α1, α2, . . . , αk, the following
identity is true:
L[α_1y_1(x) + · · · + α_ky_k(x)] = α_1L[y_1(x)] + · · · + α_kL[y_k(x)].
PROOF: For k = 2, this identity is verified by using the elementary properties of the operation of differentiation, i.e. (αy)’ = αy’ and (y1 + y2)’ = y_1’ + y_2’. Generalization for arbitrary k > 2 can be done by induction.
DEFINITION: homogeneous equation (HE)
A second-order linear equation is called homogeneous equation (HE) if its free term is zero, g(x) ≡ 0. Otherwise, it is a non-homogeneous equation (NE)
DEFINITION: linearly independent functions
Two functions y_1(x) and y_2(x) are called linearly
independent (LI) on an interval I = (a, b) if Ay_1(x) + By_2(x) ≡ 0 for all x ∈ I with constants A, B or if and only if A = B = 0. Otherwise, they are linearly dependent (LD).
DEFINITION: the Wronski determinant or Wronskian
The Wronski determinant, or Wronskian, of two functions f(x) and g(x) is the function
W(x) = Wf, g := f(x)g’(x) − f’(x)g(x)
= | f(x) g(x) |
| f’(x) g’(x) |
Note: W[f, g] = - W[f, g]
THEOREM: ABEL’S IDENTITIY: If y_1(x) and y_2(x) are two solutions of homogeneous equation
y’’ + P(x)y’ + Q(x)y = 0, then
W(x) = W[y_1(x), y_2(x)] =
C exp [-integral(P(x))dx] for some constant C.
PROOF: As y_1,2 are solutions, we have
y_1’’ + P(x)y_1’ + Q(x)y_1= 0,
y_2’’ + P(x)y_2’ + Q(x)y_2= 0.
Multiply the first equation by y2 and the second equation by y1 and subtract. This gives
y2y1’’ − y1y2’’ + P(x)(y2y1’−y1y2’) = 0;
(y2y1’’ + y2’y1’) − (y1y2’’ + y2’y1’) + P(x)(y2y1’ − y1y2’) = 0;
(y2y1’)’ − (y1y2’)’ + P(x)(y2y1’ − y1y2’) = 0;
W’ + P(x)W = 0;
dW / W = −Pdx;
W = exp[−integral(P)dx]
(typical solution)
or W = 0 (special solution), both combined in the Abel’s formula.
THEOREM: GENERAL SOLUTION OF NON-HOMOGENEOUS EQUATION: Let y_1,2 be two LI solutions of the homogeneous equation L[y] = 0,
and y_∗ is a solution of the nonhomogeneous equation L[y] = g.
Then any solution of the nonhomogeneous equation is given by y = y_∗ + C1y_1 + C2y_2 for some constants C1, C2.
(the general solution of a NE is a sum of
a particular integral (any solution of that equation) and the complementary function (general solution of the corresponding HE).)
THEOREM: SUPERPOSITION PRINCIPLES FOR NON-HOMOGENEOUS EQUATIONS: Linear combination of free terms implies similar linear combination of particular integrals.
That is, if y_j(x), j = 1, . . . , m are solution of L[y] = g_j(x), j = 1, . . . , m respectively, then
y_#(x) = m∑(j=1)c_jy_j(x)
is a solution of
L[y] = g#(x) = m∑(j=1) c_jg_j(x)
for any set of constants c_j, j = 1, . . . , m.
PROOF: By assumption, L[y_j] = g_j.
Then, by linearity of L, Lemma 2.2, we have
L[y_#] = L{m∑(j=1)cjyj(x)} =
m∑(j=1) L[yj(x)] =
m∑(j=1) gj(x) = g_# as claimed.For HE g(x) = 0 with general solution y(x) = C1y1(x) + C2y2(x) there are three cases for the complementary function:
when • m = α, β ∈ R
• m = α ∈ R
• m = p ± iq NOT∈ R
for the CF, when m = α, β ∈ R
y = C1e^αx + C2e^βx
for the CF, when m = α ∈ R repeated
y = (C1 + xC2)e^αx
for the CF, when m = p±iq NOT∈ R
y = e^px(C1cos(qx) + C2sin(qx))
STEPS FOR METHOD OF UNDETERMINED COEFFICIENTS
- FORMULATE A TRIAL SOLUTION
- SUBSTITUTE THIS INTO THE EQUATION AND EQUATE SIMILAR TERMS
- SOLVE THE SYSTEM FOR COEFFICIENTS
- THE TRIAL SOLUTION ASSIGNED TO THE COEFFICIENTS WILL GIVE THE PARTICULAR INTEGRAL
TO FIND ANSATZ: SIMILARITY RULE
g(x) is the free term
y_P(x) is the corresponding trial solution, γ is the number to be checked for resonance in the Modification Rule
SIMILARITY RULE: if g(x) = Ae^ax
y_P(x) = Be^(ax)x^k γ = a
SIMILARITY RULE: if g(x) = A_nx^n + · · · + A_0
y_P(x) = (B_nx^n + . . . B_0)x^k γ = 0
SIMILARITY RULE: if g(x) = (A_nx^n + · · · + A_0)e^ax
y_P(x) = (B_nx^n + . . . B_0)e^ax x^k γ = a
SIMILARITY RULE: if g(x) = Ccos(ωx) + Dsin(ωx)
y_P(x) = (Ecos(ωx) + Fsin(ωx))x^k γ = ±iω
SIMILARITY RULE: if g(x) = (Ccos(ωx) + Dsin(ωx))e^ax
y_P(x) = (Ecos(ωx) + Fsin(ωx))e^ax x^k γ = a ± iω
MODIFICATION RULE: If γ is distinct from root(s) of the characteristic equation
non-resonant case, we use k = 0, which is the same as to say that the Modification Rule does not apply
MODIFICATION RULE: If characteristic equation has two distinct roots and γ coincides with one of them
simple resonance and we use k = 1 in the Modification Rule
MODIFICATION RULE: If the characteristic equation has a double root, and γ coincides
with this double root
double resonance and we use k = 2 in the Modification Rule
DEFINITION: quasipolynomial (QP)
A quasipolynomial (QP) of degree n in x is a function of the form f(x) = P_n(x)e^(γx), where P_n(x) is a polynomial of degree n and γ is a constant, or a linear combination of such functions
EULER-CAUCHY EQUATIONS: three cases for roots of auxiliary equation a_2m(m−1) + a_1m + a_0 = 0
- m = α, β
- m = α double root
- m = p ± iq
EULER-CAUCHY EQUATIONS: for m = α, β
general solution is Ax^α + Bx^β
EULER-CAUCHY EQUATIONS: for m = α double root
general solution is is y = (A + B ln x)x^α
EULER-CAUCHY EQUATIONS: for m = p ± iq
general solution is is y = (Asin(q ln(x)) + Bcos(q ln(x)))x^p
