Consider the vectors x(¹)(t) = (g) and x² (1) = ( ² ).6t(a) Compute the Wronskian of x(¹) and x(²).W =(b) In what intervals are xx(1) and x(2) linearly independent?D =])u([(c) What conclusion can be drawn about coefficients in the system ofhomogeneous differential equations satisfied by x(¹) and x(²)?Choose one ▾of the coefficients of the ODE in standard formmust be discontinuous at to(d) Find the system of equations x' = P(t)x.P(t) =

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Answered: Consider the vectors x(¹) (t) = ( ) and…

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 the first order nonhomogeneous system of differential equations
Consider the following non-linear system of differential equations x' = xy - 2, y' = = X 2y. Note that (2, 1) is a stationary point of the vector field associated with the above equation. Problem 1 ● (a) A stationary point (x, y) of a vector field is called isolated if there exists a neighborhood U around (x, y) such that (x, y) be the only stationary point in U. Show that (1,2) is an isolated stationary point of the vector field associated with the above system. (b) Find a suitable change of coordinates that maps (2, 1) to (0, 0), and write the linearization of the new system. (c) Sketch the local phase portrait of the new system near the point (0,0). (d) A critical point (x, y) is called asymptotically stable whenever there exists > 0 such that the following holds if_√(x(0) − xx)² + (y(0) − y+)² < 8, Use the phase portrait that you sketched in part (c) to see whether the point (0, 0) is asymptotically stable or not. then _lim (x(t), y(t)) = (x*, Y*). t→∞
1
3. Consider the system of two differential equations given by x' = Ax, which has the [v₁ + iw₁] complex eigenvalue A = a+iß with the corresponding eigenvector v = V2+iW₂ Find the (real-valued) general solution of the system. Show all your work.
6. Sketch the vector field diagrams of the following systems of differential equations in the (x, y)-plane (a) = x – 8u ly = -x + 3y (b) x' = 2x – 4y = x – 2y (c) - 18y y = 5x – 6y 12x –
Solve the 2nd-order homogeneous Cauchy-Euler equation r²y + 23ry + 121y = 0 The linearly independent solutions are 3/2= and the general solution of the equation is y (r) = where ei, and are arbitrary constants. (Note: Let 1 and 2 be two solutions of the auxiliary (characteristic) equation. (1) If m1m2, write /1 and 1/2 = ""; ® and 2 In (r); and = (ii) If m1m2, write /1 = (iii) If m₁ = a +bi, and m₂ = a-bi, write 9/1 = cos (bln (r)) and 2 = sin (bln (z)).) symbolic formatting help
Consider a system of differential equations which has the general solution: 18]- = C, · cos (5. t) – sin (5 - t) )+C2 · sin (5 - t) + • Cos (5 - t (a) Determine the fundamental matrix (t) which has the property (0) = I. (t) = (b) Use the fundamental matrix found in part (a) to quickly determine the particular solution for the following initial conditions: r (t) : y (t) = :8]-[- (0): 1 (ii) r (t) = y (t) =
3
a) Show that the fourth-order Runge-Kutta method, k₁ = hf (ti, wi), h k₁ k₂= hfti+ Wi + 2 2 h k3 = hfti + Wi + 2¹ 2 k4 = hf (ti + h, wi + k3), 1 W₁+1 =W₁ + (k₁ + 2k2 + 2k3+k4), when applied to the differential equation y' = Ay, can be written in the form 1 Wi+1 = = (1 + hλ + 1 (hx)² + = (ha)³ + 2/7 (hX)^) wi- Wi. 24

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