(1) For the following systems of ODEs, find the general solutions (in vector form), ygen. Make...
(1) For the following system of ODES: (i) First, convert the system into a matrix equation, then, (ii) Find the eigenvalues, 11 and 12, then, (iii) Find the corresponding eigenvectors, x(1) and x(2), and finally, (iv) Give the general solution (in vector form), ygen, of the system. (Parts (i)-(iii) will be in your work) s y = -241 + 742 y2 = yı + 4y2 General Solution:
(Find the general solution of the following systems of ODES n(32 0 1 3. y (t) A y(t), Aj 0/ 4 4. y (t)A y(t)+g(t) (0-1. qlt) Aij 2cost - 8sint/ 4 Please show all steps with explanations. Thank you (Find the general solution of the following systems of ODES n(32 0 1 3. y (t) A y(t), Aj 0/ 4 4. y (t)A y(t)+g(t) (0-1. qlt) Aij 2cost - 8sint/ 4 Please show all steps with explanations. Thank you
3. Find the general solutions for the following homogeneous ODEs. dºy.dy + y = 0 a) dx2 dx d²y b) dx2 4y = 0 a) d²y dy + dx² dx = 0
Problem 15. Find the general solutions of the following linear ODES. (1) g" +3y + 2y = cos 7. (2) y" - y = sin r.
1. Find general solutions to the following differential systems of equations using dsolve: a. x' = y + t, y' = 2 -x+t b. x'=s-X, y' = -y - 3x, C. X" = x - x - y, y = -x- y - y - s', s" = -95 d. Solve the equations in c. above with the initial conditions x(0) = 1, x'(0) = 0, y(0) = -1, y'(0) = 0, $(0) = 1, s'(0) = 0, and plot...
Find the general solution by looking for solutions of the form , where r is a real or complex constant. Use the Equations of EULER-CAUCHY y(t) = + 12" + ty' +y=0,t> 0
1. Second order ODE (25 points) a. Consider the following nonhomogeneous ODEs, find their homogeneous solution, and give the form (no need to determine coefficients) of nonhomogeneous solution. (12 points) i. 44'' + 3y = 4x sin ( *2) ii. J + 2 + 3 = eº cosh(22) b. Find the general solution of y" + 2Dy' + 2D'y = 5Dº cos(Dx) where D is a real constant with following steps i) Determine homogeneous solution, ii) Find nonhomogeneous solution with...
(1 pt) Consider the following two systems. 1 3x - y 3x -y -3 () Find the inverse of the (common) coefficient matrix of the two systems. A-1 (i0 Find the solutions to the two systems by using the inverse, i.e. by evaluating A hand side (i.e. B_ Solution to system (a):x- Solution to system (b):x- B where B represents the right 2for system (a) and B- -4 for system (b) "У
2. Solve each of these ODEs using power series method expanded around Xo = 0. Find the recurrence relation and use it to find the first FOUR terms in each of the two linearly independent solution. Express your answer in general form where possible (well, it is not always possible). (a) (25 marks) (x2 + 2)y” - xy + 4y = 2x - 1-47 Note: expressa in terms of power series. (b) 2x2y" + 3xy' + (2x - 1) =...
(1 point) Find the general solution of the following system. Give the answer in real form (no complex numbers or complex exponentials) t)(t)5y(t) 2(t)-G | (2/5cost-1/5sint) y(t)-c| | cost + sint