Option a is true
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Compute the Wronskian of yı(t)= e?' and yz(t)=e-37 -5e- -e- -5e- -61 -e-61
The output of an LTI system is yı(t) when xy(t) is the input. System yı(t) -1 1 The output of the system is yz(t) when x2(t) is the input. x (t) 3 N yz(t) System Express yz(t) in terms of yı(t) using the following format yz(t) = a, y(t-by) + a2 yı(t-b2)
3. Given that yı(t) = t, y(t) = t, and yz(t) = are solutions to the homogeneous differential equation corresponding to ty" + t'y" – 2ty' + 2y = 2*, t > 0, use variation of parameters to find its general solution.
Substituting yı(t) = Coe-kat into the differential equation for yz(t) we obtain dyz = kacoe-kat – kcV2 A test solution to this differential equation takes the form yz(t) = Ae-kat + Be-kct where A and B are constants to be determined. way obtained by differentiating the test solution y(t), ay2 = -kqAe-kat – k«Be-kct Exercise: Substitute the test solution into the right hand side of the differential equation above. Show your working.
Note that yı(t) = Vt and yz(t) = t-1 are solutions of the linear homogeneous differential equation 2t’y" + 3ty' – y = 0. Use variation of parameters to find the general solution of the nonhomogeneous differential equation 2t’y" + 3ty' - y = 4t² + 4t. 8 o* Civt + Cat-1 + + 35 OB. 4 Civt + Cat-1+ t + 2 t2 9 of Civt + Cat-1 + t2 + 2t 9 00 Civt + Cut-+ 4 OE...
4. Find the Wronskian for y1 = x , y2 = cos(2x), and y3 = e . 4. (10 points) Find the Wronskian for yı = 23, y2 = cos(2x), and y3 = e3r.
Consider the vectors х() 3 (, 1)T х°() — (?, 26)Т. and i) Compute the Wronskian of X1 and X2 at t ii) In what intervals are X1 and X2 linearly independent? iii What conclusion can be drawn about the coefficients in the systems of homogeneous differential equation satisfied by X1 and X2? v) Find this system of equations and verify the conclusions of part iii Consider the vectors х() 3 (, 1)T х°() — (?, 26)Т. and i) Compute...
3t Two solutions to y'' – y' - 12y = 0 are yı = e 12 = en a) Find the Wronskian. W = Preview b) Find the solution satisfying the initial conditions y(0) = 0, y'(0) = 42 y = Preview
1. Compute the Wronskian for the following functions. Then use the Wronskian to determine whether the functions are linearly independant or linearly dependant. a) {(tan2x - sec2 x),3 (b) le,e,e) 2. Use variation of parameters to find a general solution to 2y" -4ry 6y3 1 given that y 2 and y2- 3 are linearly independant solutions of the associated homogeneous equation. (Hint: be careful the equations are in the right form.) Find a particular solution for each of the following...
(1 point) It can be shown that yı = e-4x and y2 = xe-4x are solutions to the differential equation y + 8y +16y=0 on the interval (-00, 00). Find the Wronskian of yn y (Note the order matters) W(y1, y2) = Do the functions yn y form a fundamental set on (-00,00)? Answer should be yes or no
Consider the vectors 2.0)(8) = ( = ( ),212)() = FO) (a) Compute the Wronskian of 2 (1) and 2(2) (b) On what intervals are (1) and (2) independent? (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by z(1) and 2(2)? (d) Find the system of equations t' = P(t) x and verify the conclusions of part (c).