y''-10y'+25y=30x+3

x=815y−110

Use the method of Undetermined Coefficients ...let us guess that  the solution has the form

y  = Ax + B

Then

y'  = A     and y''   = 0

So

y" - 10y' + 25y  = 30x + 3

0 - 10(A) + 25(Ax + B)  = 30x + 3

-10A + 25Ax + 25B  =  30x + 3     equate coefficients

25A = 30   →  A  = 6/5 

(-10A + 25B)  =  3    →  -10(6/5) + 25B = 3 →  -12 + 25B = 3  → 25B = 15 → B = 3/5

Then the solution  is

y  = (6/5)x + 3/5