Final answer:
To find the general solution of the differential equation y'' + 2(x-1)y' + 2y = 0 about the ordinary point x=1, we can use power series.
Step-by-step explanation:
To find the general solution of the differential equation y'' + 2(x-1)y' + 2y = 0 about the ordinary point x=1, we can use power series.
We assume that y has a power series representation of the form:
y(x) = \sum_{n=0}^{\infty} a_n(x-1)^n
By substituting the power series into the differential equation and equating coefficients of like powers of (x-1), we can solve for the coefficients a_n. The general solution will then be the sum of these power series representation terms.
For example, the coefficient a_0 can be found by substituting y into the differential equation and solving for a_0. Similarly, a_1, a_2, etc. can be found by differentiating y and substituting the resulting expression into the differential equation.
To solve the differential equation y'' + 2(x - 1)y' + 2y = 0 using power series about the ordinary point x=1, we would typically assume a solution of the form y = a0 + a1(x - 1) + a2(x - 1)^2 + ..., where a0, a1, a2, ... are constants to be determined. We then differentiate this series term by term to find y' and y'' and substitute these into the given differential equation.Through this process, each coefficient ai is determined from the recurrence relation resulting from equating the coefficients of like powers of (x - 1) to zero in the differential equation.