Higher order linear differential equations

Solution of homogeneous DE

We will investigate two cases:

1. 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
2. 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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