Linear
equations can usually be solved completely and explicitly.
Determine whether or not each equation is linear:

1.

2.

3.

4.

1.

2.

3.

4.

The work

2.

Find the general solution to the homogeneous differential equation
The solution has the form
with

Left to your own devices, you will probably write down the correct
answers, but in case you want to quibble, enter your answers so that
are normalized with their value at x=0 equal to
1.

The work

3.

Find the general solution to the homogeneous differential equation
The solution has the form
with
and Left to your own devices, you will probably write down the correct
answers, but in case you want to quibble, enter your answers so that
are normalized with their value at t=0 equal to
1.

Find the general solution to the homogeneous differential equation
The solution has the form
with
and Left to your own devices, you will probably write down the correct
answers, but in case you want to quibble, enter your answers so that
are normalized with their value at x=0 equal to
1.

Determine whether the following pairs of functions are linearly independent or not on the whole real line.

1. and

2. and

3. and

1.

2.

3.

The work

9.

Find the general solution to the homogeneous differential equation
The solution can be written in the form
with
and Left to your own devices, you will probably write down the correct
answers, but in case you want to quibble, enter your answers so that
the functions are normalized with and .