Question

4. (20) Find a second solution of zy" - 2y + y = 0 given that y = r is a solution. (a) Substitute y = (b) Compute the derivatives. Simplify your answers. Write your answers in the box (c) Substitute into DE. Show your work. Simplify your answer. Write your answer in the box.

4. (20) Find a second solution of zy" - 2y + y = 0 given that y = r is a solution. (a) Substitute y = (b) Compute the derivatives. Simplify your answers. Write your answers in the box (c) Substitute into DE. Show your work. Simplify your answer. Write your answer in the box.

Answer 1

The detailed solution is given below:

given differential equation g. x² y" - my' + y = fiven We have to find fiven that n is solution of (:). the other solution ( of (2) method of finding complele solution Herre have to use the in toruns of known integral. fa) q wue have to substitute y = golul), o is the second solution of (i) oy 19 Then no Nimi & uto no"+ aul and H2011 the above Now, putting rates into differential equations & We have - 1 2 ( Mu" + 20') 1. (x: 0'+6) + no = 0 3 он M V nvt nv - o + au'n? 3 OM/ + n 2 w 17 Ory n + (40) 01/ vi 014 d (u) old Integrating above equation logu = -lognt loge I werere loge integrating constant or va bu

Again integrating ger v (n) = clogne & d (wi where here a integrating constant So, the required se eond solution is where and & are constants u(n) = clogn logn +d de la