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Supplementary Problems

Exercise 13. If u = f(y/x) with f differentiable, show that xu + yu = 0 for
y
x
x 6= 0.

Exercise 14. a) By computing u , u , and u xy obtain a second-order partial
x
y
differential equation for u = f (x)f (y). b) Show that your result is equivalent to
1
2
(log u) xy = 0, and explain.
Exercise 15. Find partial differential equations satisfied by the following functions
u(x, y) for all arbitrary functions f and all arbitrary constants a and b: a) u(x, y) =
2
n
2
2
2
f(x − y ); b) u(x, y) = (x − a) + (y − b) ; c) u(x, y) = y f(y/x); d) u(x, y) =
f(x + ay).
Exercise 16. Find the differential equation of lowest order which possesses each
of the following solutions, with f and g arbitrary functions: a) z = (x − y)[(x + y);
b) z = f(ax + by) + g(cx + dy); (ad − bc 6= 0); c) z = f(ax + by) + xg(ax + by).

Exercise 17. a) Show that y(x, t) = F(2x+5t)+G(2x−5t) is a general solution
2
2
∂ y ∂ y
of 4 = 25 .
∂t 2 ∂x 2
b) Find a particular solution satisfying the conditions y(0, t) = y(π, t) = 0,
0
y(x, 0) = sin 2x, y (x, 0) = 0.
Exercise 18. Find general solutions for the following PDEs (u = u(x, y)) :
a) u = 1; b) u yy = 12y; c) u xy = 0; d) u xx = 6x.
x

Exercise 19. Solve the following partial differential equations for u(x, y) with
the boundary conditions given: a) x ∂u + xy = u, u = 2y on the line x = 1; b)
∂x
1 + x ∂u = xy, u(x, 0) = x.
∂y

Exercise 20. Show that each of the following equations has a solution of the
form u(x, y) = f(ax+by) for a proper choice of constants a, b. Find the constants
for each example. a) u + x + 3u = 0. b) 3u − 7u = 0. c) 2u + πu = 0.
y
y
x
y
x
Exercise 21. Show that each of the following equations has a solution of the form
u(x, y) = e αx+βy . Find the constants α, β for each example. a) u + 3u + u = 0.
y
x
b) u xx + u yy = 5e x−2y . c) u xxxx + u yyyy + 2u xxyy = 0.
Exercise 22. a) Show that there exists a unique solution for the system

(
2
u = 3x y + y,
x
3
u = x + x,
y
together with the initial condition u(0, 0) = 0.
(b) Prove that the system
(
2
u = 2.999999x y + y,
x
3
u = x + x.
y
7