PDE-17, recitations 2 and 3, First integrals. Linear
and quasilinear partial differential equations
First integrals
1. Let a linear autonomous system on the plane be of the type a) saddle b) node c)
focus d) center. Does there exist a continuous first integral of this system in some
neighborhood of zero?
2. a. Find all the continuous first integrals for a system
5 4
ẋ = Ax, A =
.
4 5
in a neighborhood of zero.
b. For which λ the system
ẋ = (A − λE)x
has continuous first integrals in some neighborhood of zero?
3. Does there exist a continuous first integral of a system
ẋ = x2 − 1, ẏ = 1 − y 2 ,
defined a) in the unite circle x2 + y 2 ≤ 1?
b) In the whole plane?
Linear equations
4. Find a general solution of the equation yux + (x − x3 )uy = 0.
5. Find a general solution of the equation yux + (x3 − x)uy = 0.
6. Is it correct that the following Cauchy problem: yux + (x3 − x)uy = 0, u|x=0 = sin y
has a unique solution in some neighborhood of zero?
7. Is the following Cauchy problem yux + (x3 − x)uy = 0, u|x=0 = sin y solvable in a
neighborhood of zero?
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8. Is the following Cauchy problem yux + (x − x3 )uy = 0, u|x=0 = cos y solvable in
a neighborhood of zero? If yes, is the solution unique in some neighborhood of the
initial line? Find the maximal domain in which the solution exists and is unique.
9. Find the general solution of the system
(x2 − y 2 )ux + 2xyuy = 0
(1)
10. Are the following Cauchy problems solvable in a neighborhood of (0, 1):
(x2 − y 2 )ux + 2xyuy = 0
a) u |x2 +y2 =1 = sin x
b) u |x2 +y2 =1 = cos x
11. Find a general solution of the problem ux + (−y + cos 2x)uy = 0
Is it correct that any closed simple curve has a characteristic point?
Quasilinear equations
12. Deduce an equation of a steady motion of particles in a no-collision media: u(x, t) is
the velocity of a particle that passes through the point x at the time t.
Answer: ut + uux = 0.
13. Solve the Cauchy problem ut + uux = 0, u|t=0 = f (x), and find the maximal segment
[a, b], a ≤ 0 ≤ b for which the solution exists for any (t, x), t ∈ [a, b]:
a) f (x) = 1;
b) f (x) = x;
c) f (x) = −x;
d) f (x) = sin x.
14. For what f from the previous problem the solution is defined on the whole (x, t)
plane?
15. Solve the Cauchy problem ut − uux = 0, u|t=0 = f (x), and find the maximal segment
[a, b], a ≤ 0 ≤ b for which the solution exists for any (t, x), t ∈ [a, b]:
a) f (x) = 1;
б) f (x) = αx.
Investigate the dependence of the answer on α.
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