The functions p(t) and g(t) are continuous for every t. It is stated that In(t+1) and...
Linear differential equations sometimes occur in which one or both of the functions p(t) and g(t) for y' + p(ty-g(t) have jump discontinuities. If to is such a point of discontinuity, then it is necessary to solve the equation separately for t < to and t>to. Afterward, the two solutions are matched so that y is continuous at to: this is accomplished by a proper choice of the arbitrary constants. The following problem illustrates this situation. Note that it is...
Let A t be a continuous family of 2 × 2 matrices and let P t) be the matrix solution to the initial value problem p for n x n matrices, but it's messier.) Show that A()P, P )-P-(The result can be proved detP(t) (detPo) exp(J0 TrA(s)ds How is this related to the Wronskian from second order differential equations? (Look back at your work on second order differential equations in mth165 or similar class, or look up the definition on...
Exercise 27.1 Are the following functionals distributions? (a) T(p) Ip(0) (b) T(p)= а, а ЕС. Σ φ(n) (0). (c) T(p) n=0 27.2 The space (IR) of test funct i. One is led naturally to require that test functions he and have bounded support. The space of nitely 9 (R) or simply 9 (recall Definition 15.1.7), est functions y differentiable is denoted by of these functions vanishes outside a bounded interval (which depends on e). (İİ) ф is infinitely differentiable in...
Differential equation
1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+4(x)-g(x) Subject to initial conditions y)oyy-Co) Consider the second order differential equation 2x2y" + 5xy, + y-r-x 2- The Existence of a Unique Solution Theorem says there will be a unique solution y(x) to the initial-value problem at x=而over any interval 1 for which the coefficient functions, ai (x) (0 S is n) and g(x) are continuous and a, (x)0. Are there any values of x for...
Problem 1 (a) If z = exp(bt), where a and b are not functions of t, then dz/dt =? (b) 2 = sin (Tī ) cos (TÊ), then the ABSOLUTE VALUE of 'Lo zdx =? Problem 2 (a) What is the most general solution to the following differential equa
5. Suppose and g are two functions continuous at x = 1. If lim and lim g(x) = 10, find f(1) and g(1). Justify your solution. mous at z = 1. If bug (in (52) s(e) * - ) = 21 6. Show that each of the following equations has a solution: a) sin(a) + x3 = 0. b) In(a)e* + 1*22 = 0).
Are the functions fi (x) = ex+4 and fz(x-er-5 linearly dependent or independent? A. Linearty dependent OB. Linearly independent Which of the following best describes the correct choice for part (a)? (Carefull) 0 A. Since the only solution to cfı + c/2 = 0 is ci = c2-0. B. Since the Wronskian equals zero for at least one x on (-o, o). C. Since the Wronskian never equals zero on (-oo, oo). D. Since the functions are scalar multiples of...
22.1 A,C
depending on whether indefinite op de ol functions and an is never zero over this interval. Additional Exercises 22.1. Find the general solution to each of the following nonhomogeneous differential equations Use variation of parameters even if another method might seem easier. For your comve- nience, each equation is accompanied by a general solution to the corresponding homoge- neous equation a. ry" - 2xy' + 2y 3x, yn = cix + c2x2 b. y + y = cot(x)...
Solve the following questions and Choose the correct answer. 1) The General solution to y" + y = 0 sty -3&y(x) = -3 y = cos(3x) + sin(-31) , 3cos(x) – 3 sin(x) 3 ) 3 Answer 2) Suppose that y(t) and y(t) are two solutions of a certain second order linear differential equation, sin(t)y" + cos(t) y' - y = 0. 0<<< What is the general form of the Wronskian Wy ) (6) ? Without solving the equation. b)...
Find the Laplace transform of each of the following functions. 1. $(t) = f*(4(t – 1)* sin(67) dt L{v(t)}(s) = b. g(t) = [ e 2-3(t-1) cos(71) dT L{$(t)}(s) = c. y(t) = e5(t-1) sin(97) cos(6(t – T)) dt L{s(t)}(s) =