Question

Partial Differential Equations

(a) Find the general solution to the given partial differential equation and (b) use it to find the solution satisfying the given initial data.

Exercise 1. 2∂u ∂x − ∂u ∂y = (x + y)u

u(x, x) = e −x 2

Exercise 2. ∂u ∂x = −(2x + y) ∂u ∂y

u(0, y) = 1 + y 2

Exercise 3. y ∂u ∂x + x ∂u ∂y = 0

u(x, 0) = x 4

Exercise 4. ∂u ∂x + 2y ∂u ∂y = e −x − u

u(0, y) = arctan y

Exercise 5. ∂u ∂x+v ∂u ∂y = −ru

(here r and v 6= 0 are constants) u(x, 0) = sin x x

Answer 1