Determine the intervals for which Theorem 1 on page 319 guarantees the existence of a solution in that interval. -- 5. 6. (a) y() – (In x)y" + xy' + 2y = cos3x (b) (x - 1)y" + (sinx)y" + Vx+4 y' + e'y = x² + 3 2. Determine whether the given functions are linearly depen- dent or linearly independent on the interval (0,-). (a) {e2, x?e2, e-*} (b) {e sin 2x, xe sin 2x, et, xet} (©) {2e21 – et, ezt + 1, e24 – 3, et + 1} 3. Show that the set of functions sinx, x sinx, x? sinx, x sinx} is linearly independent on (-0,0). 4. Find a general solution for the given differential equation. (a) y(4) + 2y" – 4y" – 2y' + 3y = 0 (b) y'"' + 3y" - 5y' + y = 0 7.
