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

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:

m(m-1) + (2 + 1/x)m + 1 + 1/x = 0

Solving this equation for m gives the roots m1 and m2.

The general solution of the differential equation is then:

y(x) = C1x^m1 + C2x^m2

Given that one solution is y1(x) = e^-x, we can substitute this into the general solution to find the constants C1 and C2.

However, the provided solution y1(x) = e^-x does not seem to fit the form of the general solution for a Cauchy-Euler equation. There might be a mistake in the problem statement.