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In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
Chebyshev's equation is the second order linear differential equation + = where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be obtained by power series:
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.
It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by e X or exp(X), is the n×n matrix given by the power series = =!
Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations. [26]
Months after its $80 million Series B fundraise, Course Hero has acquired Symbolab, an artificial intelligence-powered calculator that helps students answer and understand complex math questions.
which are a sequence of linear differential equations that can be readily solved with, for instance, Laplace transform, or Integrating factor. Finally, we can get Finally, we can get u n ( t ) = e − α n 2 π 2 L 2 t ( b n + ∫ 0 t h n ( s ) e α n 2 π 2 L 2 s d s ) . {\displaystyle u_{n}(t)=e^{-\alpha {\frac {n^{2}\pi ^{2}}{L^{2}}}t}\left ...