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Partial Differential Equations and Initial Value Problems

Please complete all the questions with a *.Question Sheet 7 1e), 2b)Question Sheet 8 1b), 2a)

Methods 3 – Question Sheet 7
HAND IN QUESTIONS MARKED *. Due Friday November 29 , 2019 by 11
am (end of the lecture).
Question 1. Give the general solutions for the following linear partial differential equations
and, in each case, find the particular solution satisfying F (s, 0) = s.
(a) Fx − Fy = F ,
(d) xFx + Fy = 0,
(b) 2Fx + 3Fy = x2 ,
(e) * yFx + xFy = F ,
(c) Fx + 5Fy = xy,
Question 2. For each of the following initial value problems:
(i) try to find the best collection of adjectives to describe the equation at hand (e.g.
inhomogeneous linear, quasilinear, linear with constant coefficients,…)
(ii) Solve the initial value problem for equations (b), (d) and (f)
(a) F Fx − Fy = y, F (s, 0) = s2 .
(d) xyFx − Fy + F 2 = 0, F (s, 0) = s.
(b) * Fx − Fy = 1, F (s, s) = s.
(e) xFx − F Fy = 1, F (s, 0) = 0.
(c) (x + F )Fx + Fy = F , F (s, 0) = s.
(f) x2 F − Fx − xFy = 0, F (s, s) = s.
Question 3.
In this question, we will prove that the Burgers equation
∂t u + u∂x u = 0
is satisfied by the velocity field of a non-viscous fluid in one dimension.
Suppose that the real line is filled with a fluid whose particles at point x are moving with
velocity u(t, x) at time t. Suppose that the particles don’t interact with one another or
experience any external force (so by Newton’s law, they have zero acceleration). Let γ(t)
be the path of one of the fluid particles so that γ̇(t) = u(t, γ(t)). Given that γ has no
acceleration, deduce that u satisfies the Burgers equation.
Question 4.
For some function G(x, y), consider the linear PDE
−y∂x F + x∂y F = G(x, y)
with the initial condition F (x, 0) = 0 for x > 0. RShow that this initial-value problem has a

single-valued solution on R2 {0} if and only if 0 G(A cos θ, A sin θ)dθ = 0 for all A. For
G(x, y) = x find this solution explicitly.
Methods 3 – Question Sheet 8
HAND IN QUESTIONS MARKED *. Due Friday December 13, 2019 by 11 am
(end of the lecture).
Question 1. Solve the initial value problems
(a) F Fx − Fy = y, F (s, 0) = s2 .
(c) xFx − F Fy = 1, F (s, 0) = 0.
(b) * (x + F )Fx + Fy = F , F (s, 0) = s.
Question 2.
Solve the initial value problems for the function φ(x, y)
(a) * φy + φ2x = 0, φ(s, 0) = s.
(b) 2φx φy − φ = 0, φ(0, s) = s2 /2.


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