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algorithm analytic antiderivative apply approach arbitrary assume Bessel equation boundary-value problem calculus Chapter characteristic equation complex compute Consider constant coefficients convergence convolution theorem defined denote derivatives eigenvalues eigenvectors equation of order error Euler equation example exponential order expressed gives graph Green's function Hence higher order Hint homogeneous equation independent eigenvectors independent solutions indicial equation initial condition initial-value problem integral interval invert Laplace transform linear algebra linear combination linear equations linearly independent mathematical method nonlinear numerical order differential equations order equations order three particular solution population power series Prob proof reader recurrence relation regular singular point result Runge-Kutta schemes second order simple singular point solution to 4-1 solve the initial-value Suppose Taylor series techniques tion values variable vector space verify Wronskian y(xo yi and y2 yi(x Yn+i zero