The general solution to the given differential equation is

    y(x) = 2.5x^2 + bx + c,


where b and c are unknown constant values.



To find these unknown values, use the given boundary conditions


    a)  at  x= 0       2.5*0^2 + b*0 + c = 100.2,   which gives  c = 100.2;


    b)  at  x= 100.1:  2.5*100.1^2 + b*100.1 + 100.2 = 100.1  which gives

                       25150.225 + 100.2b = 100.1

                       100.2b             = 100.1 - 25150.225 = -25050.125

                       b = -25050.125/100.2 = -250.0012475. 


It gives your solution/answer.