Differential Equations I

Question 1

Define a Lipschitz function

Answer 1

A function f : R2 → R, satisfies a Lipschitz condition (with Lipschitz constant L) if there exists L > 0 such that |f(x, y1) − f(x, y2)| ≤ L|y1 − y2| for all x ∈ R and all y1, y2 ∈ R.

Question 2

State Picard’s Theorem

Answer 2

Let f : R → R be a function defined on the rectangle R := {(x, y) : |x − a| ≤ h, |y − b| ≤ k} which satisfies:
P(i): f is continuous in R with |f(x, y)| ≤ M for all (x, y) ∈ R, where Mh ≤ k
P(ii): f satisfies a Lipschitz condition in R.
Then the IVP y’(x) = f(x, y(x)) with y(a) = b. has a unique solution y on the interval [a-h, a+h].

Question 3

State Gronwall’s Inequality

Answer 3

Suppose A,B ≥ 0 are constants and v is a non-negative continuous function satisfying v(x) ≤ B + A|integral from x to a of v(s)| for all x in interval [a − h1, a + h2], h1,h2 ≥ 0. Then v(x) ≤ Be^(A|x−a|) for all x ∈ [a − h1, a + h2].

Question 4

State the comparison principle

Answer 4

Let g: R2 → R be continuously differentiable and let u(x) and v(x) be differentiable functions which satisfy u’(x) ≤ g(x, u(x)) and v’(x) ≥ g(x, v(x)) on some interval I. If u(x0) ≤ v(x0) for some x0 ∈ I then we obtain that u(x) ≤ v(x) for all x ∈ I with x ≥ x0 while if u(x0) ≥ v(x0) then we obtain that u(x) ≥ v(x) for all x ∈ I with x ≤ x0.

Question 5

Define a plane autonomous system

Answer 5

A plane autonomous system of ODEs is a pair of ODEs in the form: dx/dt = X(x,y) and dy/dt = Y(x,y)

Question 6

Define a critical point of a plane autonomous system

Answer 6

A point (x0, y0) in the phase plane where X(x0, y0) = Y (x0, y0) = 0, so is a point where the solution is constant for all time.

Question 7

Define a stable critical point

Answer 7

A critical point (a, b) is stable if for all ϵ > 0 there exists δ > 0 and t0 such that sqrt[(x(t0) − a)^2 + (y(t0) − b)^2] < δ implies that sqrt[(x(t) − a)^2 + (y(t) − b)^2] < ϵ, ∀t > t0.

Question 8

Define an unstable critical point

Answer 8

A critical point (a, b) is unstable if there exists ϵ > 0 such that for all δ > 0 and t0 where sqrt[(x(t0) − a)^2 + (y(t0) − b)^2] < δ, also sqrt[(x(t) − a)^2 + (y(t) − b)^2] ≥ ϵ, ∀t > t0.

Question 9

Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where 0 < λ1 < λ2, classify the critical point.

Answer 9

Unstable node, with all curves except that on Z1 tending to Z2.

Question 10

Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1 < λ2 < 0, classify the critical point.

Answer 10

Stable node, with all curves except that on Z2 coming from the Z1 direction.

Question 11

Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1 = λ2 = λ, classify the critical point.

Answer 11

If M-λI = 0, we can write Z(t) = Ce^(λt) and it is a star, otherwise it is an inflected node. In both cases, if λ > 0, it is unstable and if λ < 0, it is stable.

Question 12

Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1 < 0 < λ2, classify the critical point.

Answer 12

Saddle.
If c1 = 0, Z(t) → 0,∞ along Z2.
If c2 = 0, Z(t) → 0,∞ along Z1.
Otherwise, Z(t) → 0 along Z1 and Z(t) → ∞ along Z2.

Question 13

Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1, λ2 are complex are their real parts are 0, classify the critical point.

Answer 13

Centre (periodic)
If the second entry of M is positive, it is clockwise, if it is negative, it is anticlockwise

Question 14

Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1, λ2 are complex are their real parts are non-zero, classify the critical point.

Answer 14

Spiral
If real part is positive, unstable
If real part is negative, stable
If the second entry of M is positive, it is clockwise, if it is negative, it is anticlockwise

Question 15

State Bendixson-Dulac’s Theorom

Answer 15

Consider the system dx/dt = X(x,y) and dy/dt = Y(x,y), with X, Y ∈ C1. If there exists a function φ(x, y) ∈ C1 with
ρ = ∂/∂x(φX) + ∂/∂y(φY ) > 0 in a simply connected region R then there can be no nontrivial closed trajectories lying entirely in R.

Question 16

Give the general form of a linear first order PDE

Answer 16

P(x, y)∂z/∂x + Q(x, y)∂z/∂y = R(x, y, z(x, y)), for some function z.

Question 17

Define a semi-linear PDE

Answer 17

A PDE is said to be semi-linear if it is linear in the highest order partial derivatives and the coefficients of the highest order partial derivatives depend only on x and y.

Question 18

Define a characteristic curve of a semi-linear PDE

Answer 18

Given the PDE P(x, y)∂z/∂x + Q(x, y)∂z/∂y = R(x, y, z(x, y)), the curve Γ = (x(t), y(t), z(t)), where x(dot) = P(x, y), y(dot) = Q(x, y) and z(dot) = R(x, y, z) is the characteristic curve.

Question 19

Define a characteristic projection

Answer 19

Given a characteristic (x(t), y(t), z(t)), the characteristic projection is the curve (x(t), y(t))

Question 20

Define Cauchy data

Answer 20

If P(x, y)y_s − Q(x, y)x_s ≠ 0 on the data curve, we say the data is Cauchy.

Question 21

Define the domain of definition

Answer 21

The region of the (x,y) plane where the solution is determined uniquely by the data.

Question 22

Give the general form of a second order semi-linear PDE

Answer 22

a(x,y)u_xx + 2b(x,y)u_xy + c(x,y)u_yy = f(x,y,u,u_x,u_y)

Question 23

Given a(x,y)u_xx + 2b(x,y)u_xy + c(x,y)u_yy = f(x,y,u,u_x,u_y), classify the PDE

Answer 23

If ac < b^2: hyperbolic
If ac > b^2: elliptic
If ac = b^2: parabolic

Question 24

Give the normal form of a hyperbolic equation

Answer 24

uφψ = G(φ, ψ, u, uφ, uψ).

Question 25

Give the normal form of an elliptic equation

Answer 25

uζζ + uηη = H(ζ, η, u, uζ , uη)

Question 26

Give the normal form of a parabolic equation

Answer 26

uψψ = G(u, φ, ψ, uφ, uψ).

Question 27

State the maximum principle for the Laplacian

Answer 27

Suppose u satisfies uxx + uyy ≥ 0, for all (x, y) in some domain D. Then u attains its maximum value on ∂D.

Question 28

State the maximum principle for the heat equation

Answer 28

Suppose that u satisfies ut − uxx ≤ 0 in a region D bounded by the lines t = 0, t = τ > 0, and two non-intersecting smooth curves C1 and C2 that are nowhere parallel to the x-axis. Then u takes its maximum value either on t = 0 or on one of the curves C1 or C2.