# Higher order linear differential equations

## Solution of homogeneous DE

We will investigate two cases:

- DE with
**non-constant coefficients** $a_0(x),\ a_1(x),\ \dots,\ a_n(x)$

Example: $(2x^2 +1 ) y'' - 4xy' + 4y = 0$ (coefficients are function of $x$) and
- DE with
**constant coefficients** $a_0,\ a_1,\ \dots,\ a_n$

Example: $y^{(4)} + 2y'' + y = 0$.

In actual practice equations of the first type have solutions, which are
usually not expressible in terms of elementary functions. And even if they are,
it is extremely difficult to find them. We will show how to find the other
solution $y_2()$ for DE of 2nd order assuming we have one solution $y_1()$ already.

If the coefficients are constant as in the second case, then the solution can be readily obtained.

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