UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 2

Teori för differentialekvationer 1998 01 19

2.1. Find all solutions to the following equations:

(a)  y \frac{∂u}{∂x} − x\frac{∂u}{∂y} = 0;

(b)  x\frac{∂u}{∂x} − y \frac{∂u}{∂y} = 0;

(c)  y \frac{∂u}{∂x} + \frac{∂u}{∂y} = 0;

(d)  y \frac{∂u}{∂x} + \frac{∂u}{∂y} = u;

(e)  y \frac{∂u}{∂x} + x\frac{∂u}{∂y} = 0;

(f)  y \frac{∂u}{∂x} + x\frac{∂u}{∂y} = x;

(g)  y \frac{∂u}{∂x} + x\frac{∂u}{∂y} = x + y.

2.2. Solve the Cauchy problem

\frac{∂u}{∂x} + (x + y)\frac{∂u}{∂y} = 1,
u(x, −x) = 0.

2.3. Solve the Cauchy problem

(1 + x^{2})\frac{∂u}{∂x} + 2xy \frac{∂u}{∂y} = 0,
u(x, x + x^{3}) = h(x).

2.4. Find a function u defined in some open neighborhood of the x-axis in R^{2} such that

x^{2} \frac{∂u}{∂x} + (y + 1)\frac{∂u}{∂y} = 0,
u(x, 0) = x.

Prove that if u is a solution to the differential equation in the whole plane, then u(x, 0) is constant for x ⩾ 0. By way of contrast, u(x, 0) need not to be constant for x < 0, but the limit lim_{x→−∞} u(x, 0) exists.