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Supplementary Problems

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There u(x, y) = xC(x, y), u = C(x, y) + xC , C(x, y) + xC − xC(x,y) = x y,
x
x
x
x
2
3
R x y x y
C(x, y) = xydx = + C (y), u(x, y) = + xC (y). The solution also has to
2 1 2 1
satisfy the additional condition (called initial condition), which we verify by plugging
in x = 1 into the general solution:
y
2
y = u(1, y) = + C (y).
1
2
y
2
So C (y) = y − , and hence
1
2
3
x y xy
2
u(x, y) = + xy − ,
2 2
which one can verify by substituting into the equation and the initial condition.

1.2 Supplementary Problems

Exercise 6. Which of these equations is linear and which is homogeneous? a)
2
2
2
∂ u + x 2 ∂u = x + y . b) y 2 ∂ u + u ∂u + x 2 ∂ u = 0.
2
2
∂x 2 ∂y ∂x 2 ∂x ∂y 2
2
2
Exercise 7. What is the order of the following equations? a) ∂ u + ∂ u = 0. b)
∂x 3 ∂y 2
2
2
4
∂ u − 2 ∂ u + ∂ u = 0.
3
∂x 2 ∂x ∂y ∂y 2
Exercise 8. Which of the following operators are linear? a) Lu = u + xu ; b)
y
√
x
2
2
Lu = u +uu ; c) L = u +u ; d) Lu = u +u +1; e) Lu = 1 + x (cos y)u +
y
x
y
x
x
x
y
u yxy − u · arctan(x/y).
Exercise 9. For each of the following equations, state the order and whether
it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. a)
2
u −u +1 = 0 b) u −u +xu = 0; c) u −u xxt +uu = 0; d) u −u +x = 0;
x
t
xx
xx
t
xx
tt
t
2 −1/2
y
2 −1/2
e) iu − u xx + u/x = 0; f) u (1 + u ) + u (1 + u ) = 0; g) u + e u = 0;
x
y
y
x
t
x
y
√
h) u + u xxxx + 1 + u = 0.
t
Exercise 10. Show that the difference of two solutions of an inhomogeneous
linear equation Lu = g with the same g is a solution of the homogeneous equation
Lu = 0.
Exercise 11. Verify that u(x, y) = f(x)g(y) is a solution of the PDE uu = u u
x y
x
for all pairs of u (differentiable) functions f and g of one variable.
Exercise 12. Verify by direct substitution that u (x, y) = sin nx sinh ny is a
n
solution of u xx + u yy = 0 for every n > 0.
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