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Chapter 1

Introduction to PDE

1.1 Solved Problems

Exercise 1. Solve the equation u xy − sin 5x + e 3y = 0 for an unknown function
u(x, y).

This isn’t too hard either. First let’s integrate in x, regarding y as fixed. So we
get
1
3y
u (x, y) = − cos 5x − e x + f(y).
y
5
Next let’s integrate in y, regarding x as fixed. We get the solution

1 1
3y
u(x, y) = − y cos 5x − e x + F(y) + G(x),
5 3
0
where F = f.

Exercise 2. Solve the equation u xy + 2u = 0.
x
We can transform the problem into an ODE by setting v = u . The new function
x
v(x, y) satisfies the equation
v + 2v = 0.
y
Treating x as a parameter, we obtain v(x, y) = C(x)e −2y . Integratingv we construct
0
the solution to the original problem: u(x, y) = D(x)e −2y + E(y), where D = C.

1
3
Exercise 3. Find a solution of the wave equation 4u − u xx = cos 2t + x .
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9
Notice that we are asked to find a solution, and not the most general solution.
We shall exploit the linearity of the wave equation. According to the superposition
principle, we can split
u = v + w,

such that v and w are solutions of

1
4v − v xx = cos 2t,
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9
1
3
4w − w xx = x .
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9
4