In Problems 13-19, find at least the first four non-zero 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,
Use the formula,
Taking derivative and substituting in the equation, we get the relation,
Hence we get the relation .
The series expansion for the function is
By expanding the series we get,
Simplify the expression.
Hence, the expression after the expansion is:
By equating the coefficients, we get,
The general solution was
Apply the initial condition and substitute the coefficient.
Hence, the first four nonzero terms are .
Show that fn(0)=0 for n=0,1,2.... and hence that the Maclaurin series for f(x) is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x0 =0 ), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.
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