Using Zill method Solve the following initial value problem

Using Zill method.

Solve the following initial value problem x^3y^m + xy^\' - y = x^2, y(1) = 1, y^\'(1) = 3, y^\"(1) = 3.

Solution

let y = x^m be the solution

of homogeneous equation

then

m(m-1)(m-2) + m - 1 = 0

(m-1) (m^2 -2m + 1) = 0

m= 1 is a solution

so general solution for homogeneous eq in this case would be

y = c_1 x + c_2 x lnx + c_3 x(ln x)^2

let the particular solution be Cx^2

then

C(2x^2 - x^2) = x^2, C=1

so x^2 is the particular sol.

hence

general solution for non homogeneous eq =

x^2 + c_1 x + c_2 x lnx + c_3 x(ln x)^2

y(1) = 1 gives c_1 = 0

y\'(1) = 3 gives c_2 = 1

y\"(1) = 3 gives c_3 = 0

So solution is x^2 + x lnx