# Chebyshev Differential Equation

## Solved Problems

### Example 1.

Find the general solution of the equation

for \(\left| x \right| \lt 1.\)

Solution.

The given equation is the Chebyshev differential equation with the fractional parameter \(n = \sqrt 2 .\) Its general solution can be written in the trigonometric form:

where \({C_1},\) \({C_2}\) are constants. Note that the solution in this case is not expressed in terms of the Chebyshev polynomials due to the irrational number \(\sqrt 2.\)

### Example 2.

Find the general solution of the differential equation

for \(\left| x \right| \lt 1.\)

Solution.

Here we deal with the Chebyshev equation with the parameter \(n = 2.\) Therefore, we can directly write the general solution in the form:

where \({C_1},\) \({C_2}\) are arbitrary constants.

We can express this solution via the Chebyshev polynomials of the first kind. As

we obtain the final answer: