Question

In each of Problems 1 through 14:

(a) Seek power series solutions of the given differential equation about the given point xo; find the recurrence relation.
(b) Find the first four terms in each of two solutions yi and y2 (unless the series terminates sooner)
(c) By evaluating the Wronskian W(Vi.y2)(ro), show that yi and y2 form a fundamental set of solutions.
(d) If possible, find the general term in each solution 8, xy" + y' + xy = 0, xo = 1

Answer 1

To solve the given differential equation 8xy" + y' + xy = 0 using power series, we assume a power series solution and find the recurrence relation, then obtain the first four terms of two solutions, evaluate the Wronskian to show they form a fundamental set, and find the general term in each solution.

Step-by-step explanation:

To solve the given differential equation 8xy" + y' + xy = 0 using power series, we follow the steps:

1. (a) Seek a power series solution: Assume the solution has the form y(x) = ∑(n=0 to ∞) a_n(x-xo)^n. Substitute this into the differential equation and equate the coefficients of like powers of x to obtain a recurrence relation.
2. (b) Find the first four terms in two solutions: Substitute the assumed power series solution into the differential equation and solve for the coefficients a_n. The first four terms will give you the particular power series solutions y1(x) and y2(x).
3. (c) Evaluate the Wronskian: Compute the Wronskian of the two solutions at the given point xo. If the Wronskian is nonzero, then the two solutions form a fundamental set of solutions.
4. (d) Find the general term in each solution: Use the recurrence relation to find the general term for each solution y1(x) and y2(x).

Answer 2

(a) For the differential equation
`\(8xy'' + y' + xy = 0\)` about the point
`\(x_0 = 1\)`, the power series solutions are sought. The recurrence relation is determined by substituting the power series into the differential equation and solving for the coefficients. (b) The first four terms in each of the two solutions
`\(y_1\) and \(y_2\)` are found by using the recurrence relation and the initial conditions. (c) The Wronskian
`\(W(y_1, y_2)(x_0)\)` is evaluated to demonstrate that
`\(y_1\) and \(y_2\)` form a fundamental set of solutions. (d) If possible, the general term in each solution is found.

Step-by-step explanation:

(a) To find power series solutions, we express y as a power series
`\(y = \sum_(n=0)^(\infty)a_n(x-1)^n\)` and substitute it into the given differential equation. Solving for the coefficients yields a recurrence relation. (b) By applying the recurrence relation and initial conditions, the first four terms in each solution
`\(y_1\) and \(y_2\)` are determined. (c) The Wronskian
`\(W(y_1, y_2)(1)\)` is calculated to establish that
`\(y_1\) and \(y_2\)` form a fundamental set of solutions. (d) If possible, the general term in each solution is found by identifying patterns in the coefficients obtained through the recurrence relation.

In summary, this systematic approach involves seeking power series solutions, determining recurrence relations, finding specific terms using initial conditions, evaluating the Wronskian, and identifying general terms if applicable.

These steps are essential in solving differential equations and establishing fundamental sets of solutions for further analysis. The use of power series allows for an analytical exploration of the solutions around the given point
`\(x_0 = 1\)`.