Question

If the functions y = x and y = xe" are linearly independent solutions of the non-homogeneous second-order linear differential equation with variable coefficients z_ yll – x(x + 2)y + (x+2)y=r, its general solution is given by Oy=C1 + C2xe" + x2 O y=C1x + C2xe - 22 None of them O y=C12 + C2z²er - 23 Oy=C12? + Cymet – x3

Answer 1

son) y=x and y=xet are linearly independent solutions a w(x,xe ²) = xex tex net 1 X x2 x = (xixel) = x²extxe* _ x ² =Bex and equation (x+2) y' + " (x+2) y By using variation of parameter Method P.I-a - 3 x dx + 2 / gx de Here g , x , Y ₂ = x et w = ex and x=x X X dx txe dx > P. I = x2 ex x²ex el = P.I = -x ² + x et ferde = ex = -x ² x X e so y = a, xtaq xe ² - x - x - Y= (a,-1)2 +0zxe* _ x² y = 0, x + pret - x² eet a,-1=c, Ecl and a cg So, option (6) is correct.