2021-11-02
Solve the given differential equation by means of a power series about the given point x0. Find the first four nonzero terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution.
RizerMix
Expert2023-04-20Added 656 answers
To solve the differential equation about the point , we can assume that the solution can be written as a power series:
where are constants to be determined.
We can differentiate y(x) with respect to x:
Substituting these expressions into the differential equation and simplifying, we get:
Grouping terms with the same power of x, we get:
To satisfy this equation for all values of x, each coefficient of must be zero. We can use this to recursively solve for the coefficients :
We also have the initial conditions and . Since y(x) is a power series, we can read off the first few nonzero terms:
Therefore, two linearly independent solutions are:
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