Note that yı(t) = Vt and yz(t) = t-1 are solutions of the linear homogeneous differential...
QUESTION 10 Note that yı(t) Vt and y(t) =t-1 are solutions of the linear homogeneous differential equation 2t²Y" + 3ty' – y=0. Use variation of parameters to find the general solution of the nonhomogeneous differential equation 2t’y" + 3ty' - y = 4t? + 4t. 8 2 OA Civt + Cat-1 + 74 35 oCivt+Cet-1 + 4 9 t2 + 2t OC Civt + Cat-1 + 4 9 t2 + 2 5 2 OD. 8 Civt + Cat-1 + t2...
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.
Suppose 01(t) and 02 (t) are both solutions to the (linear, homogeneous) second order differential equation: Y" + 3ty' + 2ty = 0. Which of the following are also solutions to the same differential equation? 0302(t) 0 g = $it) + 2^2(t) Oy=4(01(t))2 0 (01(t) + 02 (t))2
2. Consider the differential equation ty" – (t+1)y' +y = 2t2 t>0. (a) Check that yı = et and y2 = t+1 are a fundamental set of solutions to the associated homogeneous equation. (b) Find a particular solution using variation of parameters.
Consider the differential equation e24 y" – 4y +4y= t> 0. t2 (a) Find T1, T2, roots of the characteristic polynomial of the equation above. 11,12 M (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) M y2(t) = M (C) Find the Wronskian of the fundamental solutions you found in part (b). W(t) M (d) Use the fundamental solutions you found in (b) to find functions ui and Usuch...
(graded) Section 3.6: Variation of Parameters ITEMS SUMMARY Try again You have answered 1 out of 2 parts correctly. Consider the differential equation: 9ty' - 2t(t +9)y +2(t+9) y = -26, t>0. You can verify that yı = 3t and y2 = 2texp(2t/9) satisfy the corresponding homogeneous equation. a. Compute the Wronskian W between yı and 32- W(t) = b. Apply variation of parameters to find a particular solution. Bre,+2te (*),+22
2. a) (7 pnts) Solve the second order homogeneous linear differential equation y" - y = 0. b) (6 pnts) Without any solving, explain how would you change the above differential equation so that the general solution to the homogeneous equation will become c cos x + C sinx. c) (7 pnts) Solve the second order linear differential equation y" - y = 3e2x by using Variation of Parameters. 5. a) (7 pnts) Determine the general solution to the system...
Find general solutions to the nonhomogeneous Cauchy–Euler equations using variation of parameters. t2y''+3ty'+y=t-1
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.
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.