Maple 9 Learning Guide - Maplesoft

{y(x) = _C1 },7.2 Ordinary Differential Equations • 243{ ∫ y(x)1_a ln(_a) + _C1 d_a − x − _C2 = 0 }Use the explicit option to search for an explicit solution for the firstresult.> dsolve( {de2}, {y(x)}, explicit );{y(x) = _C1 },{( ∫ _Z)}1y(x) = RootOf −_f ln(_f ) + _C1 d_f + x + _C2However, in some cases, Maple may not be able to find an explicitsolution. There is also an implicit option to force answers to be returnedin implicit form.The method=laplace Option Applying Laplace transform methods todifferential equations often reduces the complexity of the problem. Thetransform maps the differential equations into algebraic equations, whichare much easier to solve. The difficulty is in the transformation of theequations to the new domain, and especially the transformation of thesolutions back.The Laplace transform method can handle linear ODEs of arbitraryorder, and some cases of linear ODEs with non-constant coefficients, providedthat Maple can find the transforms. This method can also solvesystems of coupled equations.Consider the following problem in classical dynamics. Two weightswith masses m and αm, respectively, rest on a frictionless plane joined bya spring with spring constant k. What are the trajectories of each weightif the first weight is subject to a unit step force u(t) at time t = 1? First,set up the differential equations that govern the system. Newton’s SecondLaw governs the motion of the first weight, and hence, the mass m timesthe acceleration must equal the sum of the forces that you apply to thefirst weight, including the external force u(t).> eqn1 :=> alpha*m*diff(x[1](t),t$2) = k*(x[2](t) - x[1](t)) + u(t);eqn1 := α m ( d2dt 2 x 1(t)) = k (x 2 (t) − x 1 (t)) + u(t)