Question

z1(x) = 2x3 + x ln x, z2(x) = x ln x − x3 are solutions of a second order, linear nonhomo-geneous equation L[y] = f (x).y1(x) = x−2 is a solution of the corresponding reduced equation L[y] = 0. (a) Give a fundamental set of solutions of the reduced equation L[y] = 0. (b) Give the general solution of the nonhomogeneous equation L[y] = f (x).

Answer 1

700 3 + Solution consider the second order Suncar mm hmogeneous differential equation, L[y] f Suppose that (x) 236 ex lna and Z (x) = xlnux are solutions for (1) then L[za] = f and h[22] = f 114-22] = L(21)-1(2 L(2)--[22] - f-f20 (smice defferential operalors are linear ? 7 2-Z2 is a solution for L[4]=0 (22 ² %) = 2x²+xlna- aluat x² = 3x² is a solution for (2). It is given that y(x) = x-2 is a solution for Cez. Now, surice z n and a-2 Aue limearly independent a) y ay thg GX3 + C2 (x-2) is the general solution for 6). 240,24-2} forms a fundamental set q folulim for (2) b) L(2] L[xlna +233) L[xlmX] +2 L [x3] L[alna] to (snice x is a But we know that L[xlna] (3) solution for (27.) La Pautetutar Then (3 x lux is a solution for G). Hence the general solution for G) is given by y = y + Up where up is the particulae solution for CM). * y = x²tcg (2-2) + alna

Answer 2

C1 should be x^-2, not x^3