Prove Theorem 4.2.4 and Corollary 4.2.5. Reduction of Or

Prove Theorem 4.2.4 and Corollary 4.2.5. Reduction of Order. Given one solution y1 of a second order linear homogeneous equation, y?? + p(t)y? + q(t)y = 0, (i) a systematic procedure for deriving a second solution y2 such that {y1, y2} is a fundamental set is known as the method of reduction of order. To find a second solution, assume a solution of the form y = ?(t)y1(t). Substituting y = ?(t)y1(t), y? = ?? (t)y1(t) + ?(t)y? 1(t), and y?? = ???(t)y1(t) + 2?? (t)y? 1(t) + ?(t)y?? 1 (t) in Eq. (i) and collecting terms give y1??? + (2y? 1 + py1)?? + (y?? 1 + py? 1 + qy1)? = 0. (ii) Since y1 is a solution of Eq. (i), the coefficient of ? in Eq. (ii) is zero, so that Eq. (ii) reduces to y1??? + (2y? 1 + py1)?? = 0, (iii) a first order equation for the function w=?? that can be solved either as a first order linear equation or as a separable equation. Once ?? has been found, then ? is obtained by integrating w and then y is determined from y = ?(t)y1(t). This procedure is called the method of reduction of order, because the crucial step is the solution of a first order differential equation for ?? rather than the original second order equation for y. In each of Problems 28 through 38, use the method of reduction of order to find a second solution y2 of the given differential equation such that {y1, y2} is a fundamental set of solutions on the given interval.