Find two power series solutions of the given differential equation about the ordinary point x = 0. y" + x²y' + xy = 0 Oy₁ = 1 Oy₁ = 1 Oy₁ = 1- Oy₁ = 1 - Oy₁ = 1 = 1 - 1 4 12 1 2 -x + + 2 5 672 1 4 -X + 8 5 1/2+³ +6 672 1+³+15 45 6 -X 8 14 8 and Y/₂ = x- and Y₂ = x and and Y₂ = x - Y/₂ = x - 1 3 3 1 10 1 + 1 3 -x+ + 5 1 15 + 1 5 and Y₂ = x-x² + 10 + 5 120 1 15 1 9 1 120 5 7 252 9

Find the indicated coefficients of the power series solution about 0 of the differential equation a = y=3x+ 3/2 x²+5/2 (x2x+1)y" y' - 5y = 0, y(0) = 0, y'(0) = 3 3 4 x+7/12 +-1/6 5 +0(x)

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. y=c₁ex + c₂e-x, (-∞, ∞); y" - y = 0, y(0) = 0, y'(0) = 3 y =