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Solutions to Exercises

Exercise 8. a) Linear; b) nonlinear.

Exercise 9. a) Order 2, linear inhomogeneous; c) order 3, nonlinear.

∂u 2
∂u 2
Exercise 15. a) y ∂u + x ∂u = 0; b) ( ) + ( ) = 4u; c) x ∂u + y ∂u = nu; d)
∂x ∂y ∂x ∂x ∂x ∂y
2
∂u · ∂ u , or with x and y reversed.
∂x ∂x∂y

Exercise 16. a) (x − y)(z − z ) − 2z = 0; b) bdz xx − (ad + bc)z xy + acz yy = 0.
x
y
2
2
c) b z xx − 2abz xy + a z yy = 0.

3
Exercise 18. a) u = x+f(y); b) u = 2y +yf(x)+g(x); c) u = f(x)+g(y);
3
d) u = x + xf(y) + g(y).

Exercise 19. Each equation is effectively an ordinary differential equation but

with a function of the non-integrated variable as the constant of integration; a)
y
y
−1
u = xy(2 − ln x); b) u = x (1 − e ) + xe .

Exercise 20. a) Write u = af, u = bf. Therefore a and b can be any constants
x
y
such that a + 3b = 0.

Exercise 22. a) Integrate the first equation with respect to x to get u(x, y) =
3
x y + xy + F(y), where F(y) is still undetermined. Differentiate this solution with
respect to y and compare with the equation for u to conclude that F is a constant
y
function. Finally, using the initial condition u(0, 0) = 0, obtain F(y) = 0.
b) The compatibility condition u xy = u yx does not hold. Therefore there does

not exist a function u satisfying both equations.

,
Exercise 24. Φ y u = 0.
x x

Exercise 29. sin(x − 3t/2);

Exercise 30. u(x, y) = f(y − arctan x) for any function f of one variable.

2
Exercise 32. (a) e x −y 2 ; (b) a sketch of the characteristics shows that the solution
2
2
is determined only in the region (x < y ).

101