The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, [infinity]). Find the general solution of the given nonhomogeneous equation. x2y'' + xy' + x2 - 1 4 y = x3/2; y1 = x-1/2 cos (x), y2 = x-1/2 sin (x)

y = C1x^ (-1/2) cosx + C2x^ (-1/2) sinx

Step-by-step explanation:

Given that

y1 = x^ (-1/2) cosx

y2 = x^ (-1/2) sinx

are linearly independent solutions of the nonhomogeneous equation

x²y'' + xy' + x² - 1 4y = x^ (3/2)

Then the general equation of the differential equation can be written as

y = C1y1 + C2y2

y = C1x^ (-1/2) cosx + C2x^ (-1/2) sinx