# Differential Equation 1st Order Linear Application

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first-order ordinary differential equations, symmetries and linear

. partial diﬀerential equation — whose solution gives the inﬁnitesimals of the symmetry group — has the original ﬁrst-order equation in its characteristic strip. Hence, ﬁnding these inﬁnitesimals requires solving the original equation, which in turn is.ﬀerential equation is overdetermined. Although few second or higher order equations have point symmetries, and the solution of the corresponding partial diﬀerential equation.liouvillian solutions of linear differential equations of order

. algorithm to compute Liouvillian (“closed-form”) solutions of homogeneous linear diﬀerential equations. However, there were several eﬃciency problems that made. factors into linear forms, one gets all coeﬃcients of the minimal polynomial of an algebraic solution of the Riccati equation, instead. speciﬁcally detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic (1986. decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm. c.modeling with first-order differential equations 3.1 linear

a) Radioactive Decay (half-life) eg 1 Einsteinium-253 decays at a rate proportional to the amount present. Determine the half-life > if this material loses one third of its mass in 11.7 days. Half-life is the amount of time it takes a radioactive material to lose 68a#Î$b " half of its mass. # R! œ R! /5Ð""Þ(Ñ ß 5 œ ""Þ( à # R! œ R! /5>ß $ > œ ""Þ( 68#Î68a#Î$b #! days. b) Population Growth eg 2 In 1790 the population of the US was 3.93 million, and in 1890 it .
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