3.31. The BUPy" =4 [y-x), OLXLI, y (0)=0, yYW=2has the solution y(x) = e²/04-1) ²² (e²x²_e-2x) + x.a System,itse the finite difference method to reduce the problem tot.of theform Ay =1 with h = 4.

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Answered: 3.31. The BUP y" = 4 [y-x), OLXLI, y…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve for x and y the following system of simultaneous equations 2 In(x) = In(y) + In(16) eteU = e4 subject to the restriction on x and y that x > 0, y > 0. O a. x=(-16+/512)/2, y=(24-/512)/2 O b. x=(16-/512)/2, y=(-24+J512)/2 O c. x=(-16-/512)/2, y=(-24-/512)/2 O d. Not in the list
on y=/x+3 Find the solutions (x, y) to the nonlinear system y= of Select one: O (-2, 6) O (6, 3) O (-2, 1) O (6, 3) and (-2, -1) Nor'a product was Yn = 0.25g a
Kk.295.
Solve the Attached problem
1. Solve x' = Ax + b, if A 4 -3 and b 2t -1 -1 -2 2 -18e3t 72e3t 54e3t 2. Solve x' = Ax + b, if A : 2 4 -1 and b 3 3. Find the general solution to the system d 1 2e2t x+ dt -1 4 e3t for 0 2
4. Solve the non-homogeneous systems of ODEs x' = Ax+f(t), with x(a) = xa using the given matrix exponentials. 4 -1 - ( 1 = ₂ ), !), f(t) = f(t) = ( 5 -2 (a) A = 18e2t 30e²t ), x(0) = ( 0 ), e4t. At = 1 ( -e-t + 5e³t -5e-t +5e³t e-t-e³t 5e-t e³t
2- If L(x, y) = ax + by2 is a Lyapunov function at the origin for the system * = y + x³, y = -x + y²³, a = b = 1. Then compute the expression: L(x, y) - 2(x + y4) = 3- If L(x, y) = ax² + by² is a Lyapunov function at the origin for the system x = 2x + xy², y = y(1-x²), a = b = 1, then this system is: (a) Stable (b) unstable (c) Asymptotic stable (d) stable if x 0.
3. Given the following Lyapunov candidate function, V = x"x and V = a" à + à"x = a" (A(t) + A" (t))æ 0 0 < x"x = V (t) < V(0)e¬^ asymptotically stable at the origin. *1 = -10x1 + e*x2 *2 = -2x2
6. It is known that the general solution of the system *- )* = x is given by e³t+ [¹1] = C1 Use the method of Undetermined Coefficients for systems to find the form of a particular solution to the nonhomogeneous system x': A. x₂(t) = te a+eb+tc+d (B.x₂(t) = = teta+tb+c C. xp(t) = ea + e³t b+tc+d D. xp(t) = tea+t²b+tc+d E. x₂(t) = t'e' a+e¹b+tc+d = et. x+ 4e² Le+ Xce)= (ater ) t b, It = te¹ A + TB + C IB

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3.31. The BUP y" = 4 [y-x), OLXLI, y (0)=0, y1=2 has the solution y(x) = e² (c²4_1) ~² (p²x_p-²x) + X. itse the finite difference method to reduce the problem to a system of the form Ay zb with h = t.