Question

Verify that the given function form a fundamental set of solutions on the interval (0, 0), compute the Wronskian, and form the general solution. xy'' – 6xy' +12y = 0 x?; x+ I verified the solution. ONo Yes Find the Wronskian and verify that the functions are linearly independent on the interval (0, 0). W(x", x4) = 0 Preview I found the general solution. OYes ONO

Answer 1

Given (1) x²y" – 6xy' +12y=0 - net y(x) = x3 2 9 (x ) = x4 y; (x) = 3x² - (2) y" - 6x ] From equation (1) 8 (2) x²x6x - 6xX3x² + 12x² =0 6x3 - 18*² + 12x3 =0 18x²–18x²=0 = 0 = 0 → 4,(x) is a solution of equation (1). 4 (X) = x4 2 Yg (2) = 4x3 'y (x) = 12x² /

from equation w $ (3) dhe poet m 2 (1232) x² (1282) – 6x(423) + 12x4 = 0 1284 - 24x4 + 12x4 = 0 24x4 - 2404=0 of Now => 4, (x) is a solution of equation c. rence 4, (x) = x3 and 4, (x) = x4 is a solution equation u. 1 | 23 24 W (x3, x4) = 3x2 483 = 4x²-3x6 | W (x3, x4) = x6 - x, x4 form a fundamental set of solution. therefore general solution of equation in is ginen by y(x) = 6, x² + Cg x 4