Find the general solution of each of the equation if one solution is given (i) xy′′ + (2x + 1)y′ + (x + 1)y = 0, y1(x) = e −x
Question
Solution 1
The given differential equation is a Cauchy-Euler equation. The general solution of a Cauchy-Euler equation is given by:
y(x) = x^m
where m is a root of the auxiliary equation:
am(m-1) + bm + c = 0
For the given equation, a = 1, b = 2 + 1/x, and c = 1 + 1/x. The auxiliary equation is:
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