Solve the following problem according to the method mentioned in the heading from the picture below.

2.1: problem: 3, 6, 10, 18, 22

1-6 GENERAL SOLUTION. INITIAL VALUE 17-22 Reduce to first order and solve (showing each PROBLEM step in detail). (More in the next problem set.) Verify by substitution that 17. y" = ky the given functions form a basis. Solve the given initial y" = l+y'2 18. y= value problem. (Show the details of your work.) 19. yy" = 4y' ,12 1. y" - 16y = 0, ex, e-42, y(0) = 3, y'(0) = 8 20. xy" + 2y' + xy = 0. y1 = x- cos x y" + 25y = 0, cos 5x, sin 5x, y(0) = 0.8, 21. y" + y'3 sin y = 0 y' (0) = -6.5 22. (1 - x2)y" - 2xy' + 2y = 0, y1 = x 3. y" + 2y' + 2y = 0, e-* cosx, e-sinx, y(0) = 1, y'(0) = - 1 23. (Motion) A small body moves on a straight line. Its 4. y" - 6y' + 9y = 0, e3x, xe3x, y.(0) = -1.4, velocity equals twice the reciprocal of its acceleration. y' (0) = 4.6 If at 1 = 0 the body has distance I'm from the origin 5. x2y" + xy' - 4y = 0, x2, x-2, y(1) = 11, and velocity 2 m/sec, what are its distance and velocity y' (1) = -6 after 3 sec? 6. x2y" - 7xy' + 15y = 0, x3, x5, y(1) = 0.4, 24. (Hanging cable) It can be shown that the curve y(x) y'(1) = 1.0 of an inextensible flexible homogeneous cable hanging between two fixed points is obtained by 7-14 LINEAR INDEPENDENCE AND DEPENDENCE solving y" = kV1 + y'2, where the constant & depends Are the following functions linearly independent on the given interval? on the weight. This curve is called a catenary (from 7. x, x In x (0 0) derivative is proportional to the first. 12. x - 2, x + 2 (-2 0) Solutions of Linear ODEs. Write a short essay (with 14. 0, sinh Tx (x > 0) proofs and simple examples of your own) that includes the following. REDUCTION OF ORDER is important because it gives a simpler ODE. A second-order ODE F(x, y, y', y") = 0, linear (a) The superposition principle. or not, can be reduced to first order if y does not occur (b) y = 0 is a solution of the homogeneous equation (2) (called the trivial solution)