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

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
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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:

m(m-1) + Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob

Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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