UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 1

Teori för differentialekvationer 1998 01 18

1.1. For which real numbers a does the equation

\frac{∂u}{∂t}(x, t) = (x^{2} + t^{2})^{a}

have a solution in each of the following three domains:

Ω_{1} = {(x, t) ∈ R^{2}; t > 0 or x ≠ 0},
Ω_{2} = {(x, t) ∈ R^{2}; x > 0 or t ≠ 0},
Ω_{3} = {(x, t) ∈ R^{2}; (x, t) ≠ (0, 0)}.

1.2. Study the solvability of the equation

\frac{∂u}{∂t}(x, t) = \frac{x}{x^{2} + t^{2}}

in the domain

Ω_{2} = {(x, t) ∈ R^{2}; x > 0 or t ≠ 0},

as well as in

Ω_{3} = {(x, t) ∈ R^{2}; (x, t) ≠ (0, 0)}.

1.3. Find an integral curve to the system

\frac{dx}{yz} = \frac{dy}{z} = \frac{dz}{2x − y^{2}}

which passes through the point (1, 1, 1).

1.4. Find an integral curve to the system

\frac{dx}{y − z} = \frac{dy}{z − x} = \frac{dz}{x − y}

which passes through the point (1, 1, −2).

1.5. Find an integral curve to the equation

z_{x} + 2z_{y} = 0

which passes through the curve

t ↦ (t + t^{2}, 2t^{2}, t^{2})

in (x, y, z)-space, i.e., find a function u of (x, y) such that

\frac{∂u}{∂x} + 2\frac{∂u}{∂y} = 0

and such that u(t + t^{2}, 2t^{2}) = t^{2} for t real.