In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.
The first four nonzero terms in the power series expansion of the given initial value problem is .
The power series approach is used in mathematics to find a power series solution to certain differential equations. In general, such a solution starts with an unknown power series and then plugs that solution into the differential equation to obtain a coefficient recurrence relation.
A differential equation's power series solution is a function with an infinite number of terms, each holding a different power of the dependent variable. It is generally given by the formula,
From the above equation put which is analytic over the entire number line.
Use the formula,
Taking derivative and substituting in the equation, we get the relation,
The series expansion for the function is
By expanding the series we get,
Hence the expression after the expansion is:
Expand the expression given in the previous step.
By equating the coefficients,
The general solution was
Apply the initial condition and substitute the coefficient.
Hence, the first four nonzero terms are .
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