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Prove that the general solution of the homogeneous linear system
on the interval (−∞, ∞) is
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Chapter 8 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
- Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.arrow_forwardWrite the given system as a set of scalar equations. Let x' = col (x1'(), x2 (1), x3'(1)- 0 10 1 2 x' = 0 0 1 x+t - 1 1 -11 4 .... X1 (t) = x2'(t) = X3 (t) =arrow_forwardShow that all solutions of the system - (: )- a x' : d approach zero as t → o if and only if a +d 0.arrow_forward
- 5. Let h Ah x = [h₁ + ²√5 h = √5], where h is a real parameter. (a) Find all values of h for which the system X' = AX has either a center, a spiral sink, or a spiral source. (b) Find all values of h for which X' = AX has a sink or a source.arrow_forwardProve that the system d dx(t) = −x+4y d = does not have periodic solutions by determining the values of a,n,m so that V (x,y) = xay" is a global Lyapunov function.arrow_forward
Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning