Partial Differential Equations with MATLAB

Introduction 9

Exercises 1.2

Find the general solution of each PDE. The solutionuis a function of the
variables which appear, unless otherwise stated.

1.uy=2x

2.ux=sinx+cosy

3.ux=sinx+cosy,u=u(x, y, z)

4.uyy=x^2 y

5.uxy=x−y

6.uxxy=0

7.uxxyy= sin 2x

8.uxzz=x−yz+y^3

9.uxyzz=0

10.uxyyzz=xyz

11.uy− 4 u=0,u=u(x, y)

12.ux+3u=ex,u=u(x, y)

13.ux−y^2 u=0

14.ux+3u=xy^2 +y

15.uy+xu=2

16.ux−zu=y−z

17.uxx+ux− 2 u=0,u=u(x, y)

18. Find all solutions of the PDEuy=2xwhich also satisfy the additional
    requirement that

a)u(x,0) = sinx

b) u(x,3) = sinx

c) u(0,y)=3y

19. Find all solutionsu(x, y)ofthePDEux− 2 u= 0 which also satisfy the
    additional requirement that

a)u(0,y)=y^2

b) u(1,y)=y^2

c) u(x,1) =x^2