Çözüldü Particular Solution of a Second Order Linear Differential Equation - Undetermined Coefficients Rule

Test adaptation of a solved example from the University of California - Davis:

If a particular solution for the differential equation y'' + y / x = x^2 is a cubic polynomial, what is y(1) based on that solution?

A) 4
B) 5
C) 6
D) 7
E) 8

Taking the particular solution as a cubic equation like yp(x) = Ax^3 + Bx^2 + Cx,

https://i.ibb.co/x3rkW8z/UCDavis-Dif-Eq.png
https://www.math.ucdavis.edu/~hunter/m22b_17/final_sample_solutions_22b_17.pdf
[ Page 3, Question 3.(a) ]

From yp(x), the first and second derivative functions;
y'(x) = 3Ax^2 + 2Bx + C
y''(x) = 6Ax + 2B ⇒ xy'' = 6Ax^2 + 2Bx
So the polynomial equation becomes 6Ax^2 + 2Bx + Ax^3 + Bx^2 + Cx = x^3, then organizing the left side and applying the related rule as shown above, the result is y(1) = 1 - 6 + 12 = 7