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The method of characteristics is a technique for solving for ‘quasi-linear’ partial differential equations (PDEs), i.e.

For example,

The Theory

The idea is to reduce a PDE to a series of ordinary differentiation equations (ODEs), This is achieved by forcing x, y to be linked by making them functions of one variable, t.

Comparing the PDE,

To the direction derivative

We see that

x(t) and y(t) are called the characteristic curves or simply the characteristics.

Worked Examples

Example 1: Here’s a quite easy example to get things started.

Consider the problem described thus,

given that

Firstly indentify the characteristics,

subject to

subject to

subject to

Solve these ODEs to get

(1)

(2)

(3)

Substituting (1) into (2)

(4)

Then substituting (4) into (3) gives us the solution,

Example 2: Here’s a little more tricky problem.

Consider the problem described thus,

given that

Firstly indentify the characteristics,

subject to

subject to

subject to

Solve these ODEs to get

(1)

(2)

Recall that , so,

Which can be solved using the integrating factor technique,

(3)

Substituting (1) into (2)

(4)

Then substituting (1) and (4) into (3) gives us the solution,

Example 3

Consider the problem described thus,

given that

Firstly indentify the characteristics,

subject to

subject to

subject to

Solve these ODEs to get

(1)

(2)

(3)

Substituting (1) into (3)

(4)

Then substituting (4) into (2) gives us the solution,

I hope these examples have covered most of the types of problem you’ll likely be required to solve for an exam. As usual if you have any comments use the comment box below.

Here is this post in pdf which is easier to scale.