Question

Use reduction of order and the given solution y₁(x) to solve
(1-x²) y''+2 x y'=0 ; y₁=1

Answer 1

To solve the given differential equation using the reduction of order method, substitute the second solution as y₂(x) = v(x)y₁(x), solve for v(x), and write the general solution as y(x) = c₁y₁(x) + c₂y₂(x).

Step-by-step explanation:

To solve the given differential equation (1-x²)y'' + 2xy' = 0 using the reduction of order method, we are given the first solution y₁(x) = 1. The reduction of order method assumes that the second solution can be written in the form y₂(x) = v(x)y₁(x), where v(x) is an unknown function. We substitute this into the differential equation and solve for v(x). Once we find v(x), we can find y₂(x) = v(x)y₁(x) and write the general solution as y(x) = c₁y₁(x) + c₂y₂(x), where c₁ and c₂ are constants.