214k views
4 votes
Find the general solution to the differential equation x²y'' - xy' + x = 0 and put the problem in standard form. Find the integrating factor.

1 Answer

1 vote

Final answer:

To solve the differential equation x²y'' - xy' + x = 0, we first need to convert it to standard form by dividing by x². The standard form doesn't require an integrating factor as it lacks a y term. Solutions can be sought by substituting y = xm into the standard form and solving the resulting characteristic equation.

Step-by-step explanation:

The differential equation given is x²y'' - xy' + x = 0. To find its general solution, we first need to transform it into standard form. This is done by dividing the entire equation by , assuming x ≠ 0, obtaining y'' - (1/x)y' + (1/x²) = 0, which is a second-order linear homogeneous differential equation. To find the integrating factor μ(x) for such an equation, we need it to be in the form y'' + p(x)y' + q(x)y = 0. However, since there is no y term in the original equation, the concept of an integrating factor doesn't apply directly here as it typically would for a first-order equation.

Instead, we proceed to solve the standard form equation by looking for solutions of the form y = xm. We substitute y into the standard form equation to obtain a characteristic equation in terms of m. Solving this will give us the exponents necessary to express the general solution of the differential equation.

User Abhinavkulkarni
by
8.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories