Solve the DE by Variation of Parameters Method: y" + 4y = tan 2x (2) Solve the DE by Variation of Parameters Method: y" + y = x sinx (3) Solve the Cauchy Euler DE: xy" - 4xy' + 6y = x (4) Solve the Cauchy Euler DE: xy" - xy + 2y = log x (5) Verify that the indicated function is a solution of the given PDE: Ut Uxx = 0; u(x, t) = et (cos 3x - sin 3x) (6) Find the product solution of given PDE: Uxx = cuyy (7) Evaluate: (a) L{(sint-cost)} (b) L{cos(wt+a)} (c) L{sin 2t cos 4t} (8) Evaluate: (a) L-1 L- {3 2} (6) 2-1 { 1 } 48+8 82+88+12 (9) Evaluate: (a) L{cosh at sin at} (b) L{(1-te)} (10) Evaluate: (a) L{te' cost} (b) L{t sint} (11) Use convolution theorem, evaluate L-1{2(2+1)} (12) Use Laplace transform to solve the given initial value problem: (i) y - y = 2 cost; y(0) = 0 (ii) 2y" - 2y + y = 0; (iii) y" - 6y +9y=t; 8+3 (8+1)(8+2)(8+3) y(0) = 1, y (0) = 3 y(0) = 0, y/(0) = 1